AI tutor | The No.1 Homework Finishing Free App
Numbers and Algebra
Fundamental Algebra - Algebraic Equations (Linear, Quadratic)
Q.02
'Determining coefficients from the imaginary solutions of an equation'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.03
'Let x be the amount to be repaid at the end of each year, find the value of x such that the balance at the end of each year is zero.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.05
'Find the equations of the following lines:\n(1) A line passing through the point (6,-4) and parallel to the line 3x + y - 7 = 0\n(2) A line passing through the point (-1,3) and perpendicular to the line x - 5y + 2 = 0'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.08
'Determine the types of solutions for the following quadratic equations. Where a is a constant. (1) 3x^2-5x+3=0 (2) 2x^2-(a+2)x+a-1=0 (3) x^2-(a-2)x+(9-2a)=0'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.09
'For the equation , determine the range of constant so that the equation has the following roots:'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.10
'Solve the following system of simultaneous equations.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.11
'When 0 ≤ α < π/2, sin α is the y-coordinate of point P in figure [1], and 2β (0 ≤ 2β ≤ 2π) represents the angles of radii OQ and OR.\n∠ AOQ=∠ BOP= π/2 - α, hence\n2β₁ = π/2 - α, 2β₂ = 2π - (π/2 - α)\nTherefore, β₁ = π/4 - α/2, β₂ = 3/4π + α/2\n\nWhen π/2 ≤ α ≤ π, sin α is the y-coordinate of point P in figure [2], and 2β (0 ≤ 2β ≤ 2π) represents the angles of radii OQ and OR. ∠AOQ=∠BOP=α - π/2, thus 2β₁ = α - π/2, 2β₂ = 2π - (α - π/2)\n\nTherefore, β₁ = -π/4 + α/2, β₂ = 5/4π - α/2\nFor 0 ≤ α < π/2\nα + β₁/2 + β₂/3 = α + 1/2 ( π/4 - α/2 ) + 1/3 ( 3/4π + α/2 ) = 11/12α + 3/8π\n\nHence, 3/8π ≤ α + β₁/2 + β₂/3 < 5/6π\nFor π/2 ≤ α ≤ π\nα + β₁/2 + β₂/3 = α + 1/2 ( -π/4 + α/2 ) + 1/3 ( 5/4π - α/2 ) = 13/12α + 7/24π\nTherefore, 5/6π ≤ α + β₁/2 + β₂/3 ≤ 11/8π\nFrom (1) and (2), for 0 ≤ α ≤ π, 3/8π ≤ α + β₁/2 + β₂/3 ≤ 11/8π\ny = sin(α + β₁/2 + β₂/3) is maximized\nα + β₁/2 + β₂/3 = π/2, i.e., 11/12α + 3/8π = π/2, thus α = 3/22π and the value of y at this point is 1.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.12
'Determine the types of solutions for the following quadratic equations.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.13
'Practice solving the following equations and inequalities.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.14
'Find the values of the constant for which the quadratic equation has only integer solutions, and determine all such integer solutions.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.15
'Find the sum and product of the two solutions of the following quadratic equations.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.16
'Solve the following practice problem: Find the solutions to the quadratic equation.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.17
'Important Example 23 | Solutions of Quadratic Equations and the Value of Equations Let the two solutions of the quadratic equation be , and let the two solutions of be . Here, are integers, and is a real number. (1) Express in terms of . (2) Prove that is a perfect square (can be expressed as the square of an integer).'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.18
'Calculate the probability p_{n+2} after (n+2) seconds using p_n and p_{n+1}.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.19
'(1) Let D be the discriminant of the quadratic equation x^2-k x+3 k-4=0 (1), then D=(-k)^2-4(3 k-4)=k^2-12 k+16. For the quadratic equation (1) to have complex solutions, the condition is D<0, so k^2-12 k+16<0.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.20
'Given the three lines, where a and b are constants: x-y+1=0, x-3y+5=0, ax+by=1. Prove that when these three lines pass through the same point, the three points (-1,1), (3,-1), (a, b) are collinear.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.21
'Determine the values of the constants a, b such that the equation (a-2b+4)x + (a-3b+7) = 0 becomes an identity with respect to x.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.22
'Solve trigonometric equations using sum and product formulas.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.25
'Based on the following conditions, solve the problem.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.26
'Find the equation of the line passing through the point \\( (x_{1}, y_{1}) \\) and perpendicular to the \ x \ axis.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.27
'When a cubic equation with real coefficients ax^{3}+bx^{2}+cx+d=0 has an imaginary solution α, explain about the conjugate complex numbers and demonstrate their properties.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.28
'(1) Let and be the two solutions of the quadratic equation . Determine a new quadratic equation with solutions and . (2) If and are the solutions of the quadratic equation , and one of the quadratic equations with solutions and is , find the values of the real constants and .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.29
'Find the values of α, β, γ that satisfy the following equations: \ \egin{\overlineray}{l} \\alpha^{3}=2 \\alpha^{2}+4, \eta^{3}=2 \eta^{2}+4, \\gamma^{3}=2 \\gamma^{2}+4 \\end{\overlineray} \'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.30
'Find all integer values of for which the quadratic equation has integer solutions.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.31
'Practice: Starting from the origin O on the number line, throw a coin, moving 2 units in the positive direction if it lands heads, and 481 units in the positive direction if it lands tails. Let the probability of reaching point n be denoted as pn. Here, n is a natural number.\n(1) Determine the relationship between pn, pn-1, and pn-2 for n greater than or equal to 3.\n(2) Find the value of pn.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.32
'Translate the given text into multiple languages.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.33
'(1) (6, 4) (2) In order (4x + 3y -17 = 0, 3x - 4y + 6 = 0)'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.35
'Taking the reciprocal of both sides of the recurrence relation gives \\ \\frac{1}{a_{n+1}}=4+\\frac{3}{a_{n}} \\ Letting \\ \\frac{1}{a_{n}}=b_{n} \\ we have \\ b_{n+1}=4 + 3 b_{n} \\ Rearranging this gives \\ b_{n+1} + 2=3 (b_{n}+2) \\ Also, \\ b_{1}+2 = \\frac{1}{a_{1}} + 2 = \\frac{1}{\\frac{2}{3}} + 2 = 3 Therefore, the sequence \\ \\{b_{n}+2\\} forms a geometric sequence with first term 3 and common ratio 3, where \\ b_{n}+2 = 3 \\cdot 3^{n-1} \\ which implies \\ b_{n} = 3^{n} - 2 \\ Hence, \\ a_{n} = \\frac{1}{b_{n}} = \\frac{1}{3^{n} - 2}'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.36
'(2) Let the x-coordinates of two intersection points A and B be α and β, respectively. By eliminating y from y=x^{2} and y=m(x+2), we get x^{2}-mx-2m=0. α and β are two different real solutions to this quadratic equation. Let D be the discriminant, then D=(-m)^{2}-4\\cdot 1\\cdot(-2m)=m(m+8). Since D>0, we have m(m+8)>0, which implies m<-8 and 0<m. Also, based on the relationship between the solutions and coefficients, α+β=m. Therefore, if we let the coordinates of the midpoint of line segment AB be (x, y), then x=(α+β)/2=m/2. Additionally, y=m(x+2). By eliminating m from (2) and (3), we get y=2x(x+2), which is y=2x^{2}+4x. Furthermore, from (1) and (2), we know that x<-4 and 0<x. Therefore, the sought trajectory is the part of the parabola y=2x^{2}+4x where x<-4 and 0<x.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.37
'When the equation of a circle is transformed, \\((x - 2)^2 + (y - 1)^2 = 2\\), the center of the circle C is at the point (2,1), and the radius is \\\sqrt{2}\.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.38
'Let 14k be a real number, consider the quadratic equation in x, x^{2}-kx+3k-4=0.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.39
'Consider the following conditions for integers a, b, c (*). ∫(x²+bx)dx = ∫(x²+ax)dx when integrated from a to c and b to c. (1) Express c² in terms of a, b when integers a, b, c satisfy (*) and a≠b. (2) Find all pairs of integers (a, b) that satisfy (*) and a<b when c=3. (3) Show that when integers a, b, c satisfy (*) and a≠b, c is a multiple of 3.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.40
'Find the general term of the sequence {an} defined by the following conditions.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.41
'Practice 2 curves y = 2x^{3} + 2x^{2} + a, y = x^{3} + 2x^{2} + 3x + b are tangent with the tangent line passing through the point (2,15), find the values of constants a, b and the equation of the tangent line.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.42
'Practice 39: x² = x + 3, that is x² - x - 3 = 0 has two solutions α, β (α < β), and from the relationship between the solutions and the coefficients we have α + β = 1, αβ = -3. Prove it. Also, prove that the recurrence formula is a_{n+2} - (α + β)a_{n+1} + αβa_{n} = 0. Finally, find a_{n}.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.44
'Please demonstrate that an equation with real coefficients of odd degree has at least one real solution.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.47
'(3) From the sum of two numbers α+β=-4 and the product αβ=13, find the quadratic equation and its solutions.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.48
'Classify the number of solutions of the equation sin ^{2} \\theta-\\cos \\theta+a=0 (0 ≤θ<2π) based on the value of the constant a.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.49
'Let the first term be a and the common difference be d, then the 10th term is 1 and the 16th term is 5, so a+9d=1, a+15d=5. Solving these equations gives a=-5, d=2/3. Let Sn denote the sum of the terms from the first term to the nth term. Therefore, S30=1/2*30{2*(-5)+(30-1)*2/3}=140, and S14=1/2*14{2*(-5)+(14-1)*2/3}=-28/3. Hence, S=S30-S14=140-(-28/3)=448/3'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.50
'Practice 38: Transform the recurrence relation into a_{n+2} + 4a_{n+1} = -4(a_{n+1} + 4a_{n}). Therefore, the sequence {a_{n+1} + 4a_{n}} has an initial term of a_{2} + 4a_{1} = 9, a common ratio of -4, prove it is a geometric sequence. Also, demonstrate that a_{n+1} + 4a_{n} = 9·(-4)^{n-1}. Finally, find the value of a_{n}.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.51
'For the quadratic equation with two solutions , we have and .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.52
'Let , , and be the discriminants of the three equations, respectively. Determine the range of values for a that make each discriminant have complex roots. Use the discriminant results based on the equations.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.53
'Three real numbers a, b, c form an arithmetic progression in the order a, b, c, and a geometric progression in the order b, c, a. When the product of a, b, and c is 125, find the values of a, b, and c.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.54
'From the equation of C2, we have (x-3)^2 + (y-a)^2 = a^2 - 4a + 5. Find the conditions for this equation to intersect the line y=x+1 at two distinct points.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.56
'x^{2}+y^{2}=10\n(3) y=2 x-8\n5 x^{2}-32 x+54=0\nLet the discriminant of this quadratic equation be D\nfrac{D}{4}=(-16)^{2}-5 cdot 54=-14\nSince D<0, this quadratic equation has no real solutions. Therefore, Circle (A) and Line (3) do not have any intersection points.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.58
'When a = 1, the equation for C₂ is x^2-6x+y^2-2y+8=0. Now, let k be a constant and consider the following equation: k(x^2+y^2-4)+x^2-6x+y^2-2y+8=0. Find the conditions for this to form a line.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.59
'The condition for both solutions to be greater than 4 is D>0 and (α-4)+ (β-4)>0 and (α-4)(β-4)>0'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.60
'Since the midpoint of the line segment PQ is (3+p)/2, (4+q)/2 lies on the line ℓ, therefore'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.62
'Solve the following 4th degree equation: x^{4}=4'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.63
'Prove that at least one of \ a, b, c \ is 1 when \ a+b+c=1, \\frac{1}{a}+\\frac{1}{b}+\\frac{1}{c}=1 \.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.64
'Please explain the solutions to the equation (x-3)^{2}(x+2)=0 and its repeated root.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.65
'Let be a quadratic polynomial. The polynomial cannot divide , but {}^2 can be divided by . Prove that the quadratic equation has a repeated root.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.66
'Since the real number is positive, it follows that and . Therefore, . And since .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.67
'Interpret this as solving for b in the 2nd degree equation'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.68
'Using the relationship between the roots and coefficients, find the following value.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.69
'Important Example 27 | Solutions of Two Equations'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.70
'For vegetable A, each contains 8g of nutrient x₁, 4g of nutrient x₂, and 2g of nutrient x₃, while for vegetable B, each contains 4g of nutrient x₁, 6g of nutrient x₂, and 6g of nutrient x₃. Selecting some of each of these two types of vegetables to mix and make vegetable juice. The goal is to have nutrient x₁ at least 42g, nutrient x₂ at least 48g, and nutrient x₃ at least 30g in the selected vegetables. When making juice with as few vegetables of type A and B as possible, the combination of the number of vegetable A, a, and vegetable B, b, is'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.71
'For all natural numbers n, derive cn + 1 by utilizing an + bn + cn = 1.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.72
'Lake 37 book p. 119 Finding the equation of the circle in the form x^2+y^2+lx+my+n=0. The circle passes through point A(8,5), so 8^2+5^2+8l+5m+n=0; passes through point B(1,-2), so 1^2+(-2)^2+l-2m+n=0; passes through point C(9,2), so 9^2+2^2+9l+2m+n=0. Simplifying gives 8l+5m+n=-89, l-2m+n=-5, 9l+2m+n=-85. Solving these equations gives l=-8, m=-4, n=-5. Therefore, the required equation is x^2+y^2-8x-4y-5=0. Another approach is that the circumcenter of triangle ABC is the center of the desired circle. The equation of the perpendicular bisector of AB is y-3/2=-1(x-9/2), hence y=-x+6. It can also be verified by substituting x=y=0 into 4(x+5)^2+(y-4)^2=r^2. From (1)-(2) ÷ 7 we get l+m=-12, from (1)-(3) we get l-3m=-4, thus 4m=-16, etc.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.73
'Example 4 | Three numbers forming an arithmetic progression\nThere are three numbers forming an arithmetic progression, with a sum of 18 and a product of 162. Find these three numbers.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.74
'Let the volume of a rectangular parallelepiped be denoted by V, where V = x y z is derived from equations (2), (3), (4), x, y, z are the roots of the cubic equation t^3 - 5 t^2 + 8 t - V = 0. The condition for the existence of positive numbers x, y, z is that equation (5) has three positive roots.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.75
'Example 42 | Equation of a Line Passing Through a Fixed Point\nLet k be a constant. The line (2k+1)x+(k-4)y-7k+1=0 passes through a fixed point regardless of the value of k. The coordinates of that fixed point are denoted by A□. Also, when the slope of this line is 1/3, the value of k is represented by B□.\n[Fukuoka University]'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.76
'a³ - a² - b = 0 or 9a + 27b - 1 = 0 where a ≠ 1/3'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.77
'The base is a positive number that is not equal to 1.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.78
'Find the condition for one of the equations to have complex roots.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.80
'Find the quadratic equation using the sum and product of two numbers.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.81
'Example 18 The value of the symmetric equation (2)\nFor the two roots of the 2nd degree equation , find the values of the following expressions.\n(1) \n(2) '
A. ...
Ask AI tutor for answer!
Join our DiscordQ.82
'Example 38 Recurrence Relation Among Adjacent 3 Elements (2)'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.83
'Show the solutions and discriminant of the quadratic equation ax² + bx + c = 0 with real coefficients.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.84
'Find the equation when the line passing through the intersection points of 2x - y - 1 = 0 and x + 5y - 17 = 0 becomes parallel to 4x + 3y - 6 = 0.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.85
'(1) Find the quadratic equation from the sum of two numbers α+β=7 and the product αβ=3, and solve for the roots.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.87
'(1) Find the roots of a quadratic equation using the sum and product of two numbers.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.88
'Example 17 | Value of Symmetric Expressions (1)\nSecond-degree equation x^{2}+3x+4=0\n(1) \\alpha^{2}\eta+\\alpha\eta^{2}\n(4) \\alpha^{3}+\eta^{3}\nLet the two solutions of the equation be \\alpha, \eta, then find the values of the following expressions.\n(2) \\alpha^{2}+\eta^{2}\n(3) (\\alpha-\eta)^{2}\n(5) \\frac{\eta}{\\alpha}+\\frac{\\alpha}{\eta}\n(6) \\frac{\eta}{\\alpha-1}+\\frac{\\alpha}{\eta-1}'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.89
'In (2), the relationship between the roots and coefficients is α+β=-p and αβ=q. In x²+qx+p=0, the relationship between the roots and coefficients is α(β-2)+β(α-2)=-q, α(β-2)+β(α-2)=p, and 2αβ-2(α+β)=-q. Therefore, 2q+2p=-q, which implies 2p+3q=0. From (2), we get αβ+αβ-2(α+β)+4=p, and from (1), we get q(q+2p+4)=p, thus p=-3/2q. Substituting (6) into (5) and simplifying, we get 4q²-11q=0, which leads to q(4q-11)=0. Solving this gives q=0 and 11/4. When q=0, from (6) we find p=0. In this case, α=0 and β=0, which contradicts the assumption that α and β are not equal. When q=11/4, from (6) we find p=-33/8.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.90
'Let the discriminants of the equations (1) and (2) be denoted as D1 and D2.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.92
'Since the point (1,2) lies on the line (3), we have a+2b=1'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.93
'Find the range of possible values for y to satisfy y=-2x+3 for x within the range -3 ≤ x ≤ 2.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.94
'Prove the following equation:\n\na^3 + b^3 + c^3 = -3(a + b)(b + c)(c + a) \nwhere a + b + c = 0.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.95
'Show the relationship between the solutions of a cubic equation and the coefficients.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.96
'Determine the value of the constant k that satisfies the following conditions:\n(1) One solution is twice the other solution\n(2) One solution is squared of the other solution'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.97
'For real numbers a, b, let f(x) = x^3 - 3ax + b. Let M be the maximum value of |f(x)| for -1≤x≤1.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.98
'Let the coordinates of point P be (a, b). The x-coordinate of the points where the line with slope m passing through point P intersects curve C is the real solution of the equation x^3 - x = m(x-a) + b. When this equation has three distinct real solutions, the line ℓ intersects curve C at three distinct points.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.00
'Find the value of k that satisfies the following conditions.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.01
'Find the equation of the line passing through two distinct points \\( (x_{1}, y_{1}), (x_{2}, y_{2}) \\).'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.02
'Assuming the given sequence is an arithmetic progression with the first term as 5 and common difference as -7. If the nth term of this arithmetic progression is -1010, then 5+(n-1)×(-7)=-1010. Solving this equation gives 7n=1022, which means n=146 (a natural number). Therefore, the given sequence can be an arithmetic progression. Additionally, -1010 is the 146th term.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.03
'Given mathematical text converted to multiple languages.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.04
'Practice 39⇒This book p.91\\ From the relationship between the solutions and coefficients of a cubic equation \\ α+β+γ=2, \\αβ+βγ+γα=0, αβγ=4\\'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.05
'When the equation represents a circle\n(1) Find the range of values for the constant .\n(2) When varies within this range, find the trajectory of the center of the circle.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.06
'For all values of x, y, and z that satisfy x-2y+z=4 and 2x+y-3z=-7, the constants a, b, and c need to be determined such that ax^2+2by^2+3cz^2=18 holds true.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.07
'Find the values of constants a, b, and c that satisfy the equations x - 2y + z = 4 and 2x + y - 3z = -7 for all values of x, y, and z that satisfy those equations.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.08
'Determine the values of constants a, b, c so that the equation 3x^2-2x-1=a(x+1)^2+b(x+1)+c is an identity in terms of x.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.09
'Find the number of distinct real solutions to the following cubic equations.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.10
'Confirm logarithmic equations and conditions of exponents'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.11
'When the cubic equation has a double root, find the value of the constant .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.13
'Find the two numbers that have the sum and product as follows:'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.14
'Develop 52: Proof problem regarding the solutions of a quadratic equation'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.15
'Determine the types of solutions for the following quadratic equations. Note that k in (4) is a constant.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.16
'Find the conditions under which the given polynomial P(x) = 5x^3 - 4x^2 + ax - 2 is divisible by x = 2 and x = -1.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.17
'Determine the types of solutions for the following quadratic equations. Note that k in (4) is a constant. (1) x^{2}-5x+3=0 (2) 4x^{2}+28x+49=0 (3) 13x^{2}-12x+3=0 (4) x^{2}+6x+3k=0'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.18
'When the quadratic equation has two different solutions both greater than 1, find the range of values for the constant .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.19
'For the quadratic equation with two solutions and discriminant :\n1. are two distinct positive solutions and and \n2. are two distinct negative solutions and and \n3. are solutions with opposite signs '
A. ...
Ask AI tutor for answer!
Join our DiscordQ.20
'Consider the signs of the differences \\\alpha-k, \eta-k\ of the real roots \\\alpha, \eta\ of a quadratic equation and a real number \k\\n\nFocus on the signs of the sum \\( (\\alpha-k)+(\eta-k) \\) and the product \\( (\\alpha-k)(\eta-k) \\)'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.21
'Find the number of distinct real solutions for the following cubic equations.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.22
'Determine the range of values for the constant such that the quadratic equation has two distinct real roots both greater than 1.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.23
'Determine the range of values for the constant m so that the quadratic equation satisfies the following conditions: (1) Has two positive roots. (2) Has two different negative roots. (3) Has roots with opposite signs.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.24
'Let a, b be constants. Find the values of a and b when the polynomial x^3-x^2+ax+b is divisible by the polynomial x^2+x+1.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.25
'Find the first term and the common ratio of a geometric series such that the sum of the first three terms is -7 and the sum of terms from the third to the fifth is -63.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.26
'Higher degree equation: Find the value of the constant and the other root of the equation , given that one of the roots is .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.28
'For the quadratic equation , answer the following questions.\n(1) Determine the range of values for the constant when the equation has two complex solutions.\n(2) Find the values of the constant and the repeated root when the equation has a repeated root.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.29
'Let α and β be the two solutions of the quadratic equation x^{2}-3x+4=0. Find the values of the following expressions:'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.30
'Show the formula for solving the quadratic equation ax^2 + bx + c = 0 and find its roots.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.31
'Solve the following equations for 0 ≤ θ < 2π: (1) 2cos²θ - √3sinθ + 1 = 0 (2) 2sin²θ + cosθ - 2 = 0'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.32
'Determine the values of the constants a, b, and c such that the equation x^2+2x-1=a(x+3)^2+b(x+3)+c holds as an identity with respect to x.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.33
'If the three solutions of the cubic equation are denoted as , find the values of and .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.34
'When the cubic equation has a repeated root, find the value of the constant .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.35
'Find the sum and product of the two solutions to the following quadratic equations.\n(1) \n(2)\n(3) '
A. ...
Ask AI tutor for answer!
Join our DiscordQ.37
'Basic 62: Solving Higher Degree Equations (2) - Utilizing the Factor Theorem'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.38
'Solution: Using the formula x = (-b ± √(b^2 - 4ac)) / 2a, where a = 1, b = -3, c = -3. Answer: x = 3 or x = -1.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.40
'Determine the types of solutions for the following quadratic equations. Here, k in equation (4) is a constant.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.43
'Show the relationship between the solutions of a quadratic equation and its coefficients. Let the solutions of the quadratic equation be α and β for ax^2+bx+c=0, then using the formula for solutions, demonstrate the following relationships:\n\n1. Sum of solutions α+β\n2. Product of solutions αβ'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.44
'Find the general term of the sequence defined by the recurrence relation .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.45
'When S_{2}=2 S_{1} \\quad-\\frac{1}{6}(m+3)^{3}=9 \\text{, that means} \\quad(m+3)^{3}=-54 \\text{, since } m \\text{ is a real number} \\quad m=-3-3 \\sqrt[3]{2}'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.46
'If the two solutions of the quadratic equation x^2 + 2x - 4 = 0 are α and β, then what is the quadratic equation with solutions α + 2 and β + 2?'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.47
'Find the values of x and y when the identity (k-1) x + (3-2k) y + 4k-7 = 0 holds for all values of k.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.48
'Find the range of values for the constant when the quadratic equation has two distinct solutions, both of which are less than 3.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.49
'Extension 53: Integer solutions of quadratic equations (using the relationship between solutions and coefficients)'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.50
'Solve the high degree equation x^{3}-4 x^{2}+2 x+4=0.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.51
'Basic Example 62 Determine Coefficients of Degree 64 Polynomial (1) ... Conditions for Real Solutions 3rd degree equation has -1 and -3 as solutions. (1) Find the values of constants and . (2) Find the other solutions to this equation.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.52
'When the quadratic equation has two distinct positive solutions, find the range of values for the constant .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.53
'Consider the polynomial P(x)=x^{3}-2 x^{2}+qx+2r. When the solutions of the cubic equation P(x)=0 are -2 and two natural numbers α, β(α<β), find the values of α, β, q, and r.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.54
'When the quadratic equation x^2+2mx+15=0 has the following roots, find the value of the constant m and the two roots.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.55
'Find the formula for the solutions of the quadratic equation ax^{2}+bx+c=0.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.56
'Find the value of the constant and the two solutions when the two solutions of the quadratic equation satisfy the following conditions:'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.57
'Find the general term of the sequence {an} determined by the 25 conditions. (1) a1=1, an+1=2an-3 (2) a1=1, 2an+1-an+2=0'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.59
"Find information about 'high-degree equations' based on the following table."
A. ...
Ask AI tutor for answer!
Join our DiscordQ.62
'Find the value of the constant m and the two solutions of the quadratic equation such that it satisfies the following conditions: (1) One solution is three times the other. (2) The ratio of the two solutions is 2:3.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.64
'Find the equation of a circle passing through three points.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.65
'Development 69: Solution of Higher Order Equations (3)'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.67
'Proving the equation A=B in 3 ways\n\nThe equation A=B may have conditions attached to it, but basically it is an identity. There are three styles of proving the equation as follows:\n\n(1) Comparing both sides, transforming the more complex side to derive the simpler side.\n\nA=⋯⋯ Transformation ⋯⋯ = B\n(or B =⋯⋯ Transformation ⋯⋯ = A)\n\nTherefore A = B\n\n(2) Transforming both sides separately to obtain the same expression C.\n\nA=⋯⋯ Transformation ⋯⋯ = C\n\nB =⋯⋯ Transformation ⋯⋯ = C\n\nTherefore A = B\n\n(3) Transforming A - B to show A - B = 0.\n\nA - B =⋯⋯ Transformation ⋯⋯ = 0\n\nTherefore A = B'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.68
'Let TR be real numbers, and let the equation x ^ {3}-2 x ^ {2} + ax + b = 0 have x = 2 + i as a root. Find the values of a, b and all the roots of the equation.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.69
'Find the range of values for the constant when the equation has real solutions.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.70
'Find the equations of the following lines:\n(1) Passing through the point (3, 0) with a slope of 2\n(2) Passing through the point (-1, 4) with a slope of -3\n(3) Passing through the point (3, 2) and perpendicular to the x-axis\n(4) Passing through the point (1, -2) and parallel to the x-axis'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.71
'The function takes the minimum value at . Find the value of .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.72
'46 (1) 6x^2 + x - 12 = 0 (2) 4x^2 - 12x + 7 = 0 (3) 3x^2 - 4x + 3 = 0'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.73
'Determine the values of constants a, b, and c such that the following equation is an identity in x: (1) (a+b-3) x^{2} + (2a-b) x + 3b - c = 0'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.74
'Investigate the conditions for a cubic equation to have repeated roots'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.75
'Substitute the third equation into the first equation to obtain the following equation: a^{2} + (-7a + 25)^{2} = 25. Simplifying, we get the following quadratic equation: a^{2} - 7a + 12 = 0. Therefore, we get the following solutions: (a - 3)(a - 4) = 0, so a = 3, 4. Substituting these values into the third equation, we get the following: when a = 3, b = 4; when a = 4, b = -3. Thus, the equations of the tangents are as follows: 3x + 4y = 25, 4x - 3y = 25'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.76
'Basic 43: Value of two solutions of a symmetric equation'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.77
'When the maximum value of the function f(x) = a x^3 + 3 a x^2 + b(-1 ≤ x ≤ 2) is 10, and the minimum value is -10, find the values of constants a, b.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.78
'When S_{2}=2 S_{1}, \\frac{1}{6}(m+3)^{3}=9, that is (m+3)^{3}=54. Since m is a real number, m=-3+3 \\sqrt[3]{2}'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.79
'When k=0, there is one real solution; when k=-1, there is a repeated root; when -1<k<0 or 0<k, there are two distinct real solutions; when k<-1, there are two distinct imaginary solutions.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.80
'Basic 42: Sum and product of the two solutions to a quadratic equation'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.81
'Let k be a constant. Determine the types of solutions of the equation kx^2 + 4x - 4 = 0.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.82
'Find the values of m such that the lines l1 and l2 are parallel or perpendicular.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.83
'Find the number of distinct real solutions to the following cubic equations:\n(1) -x^{3}+3x^{2}-1=0\n(2) x^{3}-3x^{2}+3x+1=0'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.84
'Find the values of x and y such that (k+2)x-(1-k)y-k-5=0 holds for any value of k.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.86
'The condition for having only imaginary solutions is'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.87
'Find the solutions of the quadratic equation x^2=k. Here, k is any real number.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.88
'Let the first term of a geometric sequence, with a common ratio r, be a, with the second term being 4 and the sum of the terms from the first to the third being 21. Therefore, we have a= and common ratio r=.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.89
"Considering q, r as real numbers, let's look at the polynomial P(x)=x^{3}-2 x^{2}+q x+2 r. If the solutions of the 333rd equation P(x)=0 are -2 and two natural numbers \\( \\alpha, \eta(\\alpha<\eta) \\), find \ \\alpha, \eta \ and \ q, r \. [Similar to center test]"
A. ...
Ask AI tutor for answer!
Join our DiscordQ.92
'Development 54: Range of existence of solutions of a quadratic equation (2)'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.94
'A and B solved the same quadratic equation in terms of x. A mistakenly obtained the coefficient of x² as 26-2/3, with a solution of 1. B mistakenly obtained the constant term as -1/3, with a solution of 1/2. Find the solutions to the original correct quadratic equation.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.95
'Find the range of values for the constant when the cubic equation has three distinct real roots.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.97
'Standard 65: Determining coefficients of higher order equations (2) - Conditions for imaginary solutions'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.98
'Find the value of the constant when the cubic equation has a double root.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.99
'Standard 49: Range of existence of solutions of a quadratic equation (1)'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.00
'Find the first term and common ratio of a geometric sequence such that the sum from the 3rd term to the 5th term is -63 and the sum from the 1st term to the 3rd term is -7.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.01
'Basic 41: Conditions for a quadratic equation to have complex roots, repeated roots'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.02
'Let the three solutions of the cubic equation be . Find the value of the following expressions.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.03
'Extension Study - Development 192 The number of real solutions of a cubic equation (3) Utilizing extreme values'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.04
'58 divided by, in order of remainders (1) x^2+2x-6, -10 (2) x^2-5x+4, 3'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.05
'Find the general term of the sequence \ \\left\\{a_{n}\\right\\} \ that represents the sum \ S_{n} \ from the first term to the nth term.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.06
"When the function f(x) represented by a polynomial satisfies f'(x)-f(x)=x²+1, f(x) is a □ degree function, and f(x)= ."
A. ...
Ask AI tutor for answer!
Join our DiscordQ.07
'Standard 40: Discrimination of types of solutions of quadratic equations (2)'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.08
'By rearranging the equation x-2y+6=0, we can express it as y=\\frac{1}{2}x+3, thus representing a line with a slope of \\frac{1}{2} and a y-intercept of 3.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.09
"Determine the range of the constant 'm' so that the quadratic equation satisfies the following conditions: (1) has two positive roots, (2) has two distinct negative roots, (3) has roots of different signs."
A. ...
Ask AI tutor for answer!
Join our DiscordQ.10
'Expansion 66: Relationship between the solutions of a cubic equation and its coefficients'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.11
'Chapter 3 High Degree Equations - 49\nEX Let a, b, c, d be real constants. The polynomial P(x) = ax^{3} + bx^{2} + cx + d leaves a remainder of 29 when divided by x^{2}-1 and a remainder of 3x + 4 when divided by x^{2}+1. In this case, a=->□, b=-1□, c=d=√□. [Shonan University]\nLet Q(x) be the quotient when P(x) is divided by x^{2}-1, and let R(x) be the quotient when P(x) is divided by x^{2}+1. Then, the following equations hold.\n\nP(x) = (x+1)(x-1)Q(x) + x + 2\nP(x) = (x^{2}+1)R(x) + 3x + 4\nP(1) = 3, P(-1) = 1, P(i) = 4 + 3i'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.13
'Find the equation of the line passing through two different points (x1, y1) and (x2, y2).'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.14
'Find the sum and product of the two roots of the following quadratic equations.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.16
'Find two numbers whose sum is 2 and product is -4.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.17
'Find the range of values for the constant when the cubic equation has three distinct real roots.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.18
'Developing 68: Conditions for a cubic equation to have three distinct real roots'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.19
'Prove that the equation a^{2}+b^{2}=c^{2}-2 a b holds true when a+b+c=0.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.20
'When the equation \\(a\\left(x^{2}-x+1\\right)=1+2x-2x^{2}\\) has real solutions, find the range of values for the constant \a\.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.21
'Regarding the underlined part g, last year the Ministry of Land, Infrastructure, Transport and Tourism also conducted a subsidy grant solicitation aimed at promoting the adoption of next-generation vehicles. Choose the correct combination of the following statements X・Y regarding next-generation vehicles as either true or false.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.22
'Choose the appropriate expression to represent the distance moved by block A with respect to block C, and provide the symbol.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.23
'1 (1) \y=mx-2m+2 \\n(2) \u=\\frac{m-1}{m}, v=1-m \\n(3) \y=\\frac{1}{x-1}+1 \, Figure omitted'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.24
'Prove that the following equations have at least one real number solution in the given range.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.25
'Let a be a real number, find the number of real solutions of the equation f(g(x))+f(x)-|f(g(x))-f(x)|=a.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.26
'Please remove the denominator and solve the following equation:\n(2x-3)(x^{2}-3x+1)=0'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.27
'Find a fifth-degree polynomial f(x) that satisfies conditions (A) and (B) simultaneously.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.28
'Let a, b be real numbers, and suppose that the cubic equation x^3+ax^2+bx+1=0 has an imaginary root α. Show that the conjugate complex number of α, denoted by α¯, is also a root of this equation. Express the third root β and the coefficients a, b in terms of α and α¯.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.29
'Find the velocity, acceleration, position, and distance traveled (linear motion).'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.30
'Prove that when a > 1, the two solutions of the equation a x^2 − 2 x + a = 0 (1) are denoted as α and β, and the two solutions of the equation x^2 − 2 a x + 1 = 0 (2) are denoted as γ and δ. Let A(α), B(β), C(γ), D(δ) and prove that the four points A, B, C, D lie on a common circle.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.31
'Fundamentals 8: Algebraic solutions for irrational equations and inequalities'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.32
'For a point on the hyperbola with a tangent line having a slope , answer the following questions. Assume .\n(1) Find the relationship between .\n(2) Let the distance between a point on this hyperbola and the line be denoted as . Find the minimum value of . Also, determine the coordinates of the point on the curve that provides the minimum value of .[Kanagawa University]'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.33
'What geometric shape is formed by the set of points that satisfy the following equations?'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.34
'Solve the equation \ \\frac{1}{x} + \\frac{1}{x-1} + \\frac{1}{x-2} + \\frac{1}{x-3} = 0 \.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.35
'20 (1) \ |\\alpha|^{2} \\n(2) Omitted (3) Maximum value when \ a=b \ is \ \\frac{1}{2} ; a=1, \\quad b=3 \ and minimum value is \ \\frac{3}{10} \'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.37
'Let two complex numbers w and z (z ≠ 2) satisfy w = iz/(z-2).\n[Hirosaki University]\n(1) When point z moves on the circumference of a circle with radius 2 centered at the origin, what shape does point w trace out?\n(2) When point z moves on the imaginary axis, what shape does point w trace out?\n(3) When point w moves on the real axis, what shape does point z trace out?'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.38
'Radioactive substances like radium decrease in mass at a rate proportional to the mass at each instant. Express the mass x as a function of time t with the proportionality constant k (k > 0) and initial mass A. Additionally, for radium, it takes 1600 years for the mass to halve. Approximately what percentage of the initial amount is left after 800 years? Round to the nearest whole number.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.40
'Solve the inequality \ \\log _{2} 256 x > 3 \\log _{2 x} x \. Let \\\log _{2} x = a \.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.42
'Consider complex numbers z that satisfy conditions (A) and (B) simultaneously. (A) z + i/z is real (B) The imaginary part of z is positive. (1) Let |z|=r, express z in terms of r. (2) Find the z for which the imaginary part of z is maximum.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.43
'Let a ≠ 0. For the function f(x) = 2ax - 5a^2, find the value of the constant a such that f^{-1}(x) and f(x) are equal.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.44
'Assuming the existence of a sequence {a_{n}} and its sum from the first term to the nth term'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.45
'Solve the following quadratic equations:\n(1) \n(2) \n(3) \n(4) \n(5) \n(6) '
A. ...
Ask AI tutor for answer!
Join our DiscordQ.46
'Prove the relationship between the roots and coefficients of the following quadratic equation. For a quadratic equation ax² + bx + c = 0, let the two roots be α and β. Then, α + β = -b/a and αβ = c/a.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.47
'As a sprinter in the 100m race, Tarou decided to focus on (1) and figure out the best stride and pitch to improve his time.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.49
'Find the range of values for the constant a that satisfy the given conditions for the two quadratic equations and .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.50
"To determine the range in which the solutions of a quadratic equation exist, let's consider the graph that satisfies the following conditions:"
A. ...
Ask AI tutor for answer!
Join our DiscordQ.51
'Let a and p be constants. Find the real solutions of the following equations in x.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.52
'Solve the following system of simultaneous equations.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.53
'How many ways are there to divide 12 different books as follows?'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.54
'(Example) In the case of 2x + 3y = 33 (where x and y are natural numbers), using y ≥ 1, we have 2x = 3(11 - y) ≤ 30, hence 1 ≤ x ≤ 15.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.55
'If the two distinct real solutions of the quadratic equation are denoted as and , and satisfy , determine the range of values of the constant .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.56
'When the equation has two solutions , find the values of constants .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.57
'For the quadratic equation \ x^{2}-a^{2} x-4 a+2=0 \ with two distinct real solutions \ \\alpha, \eta \ where \ 1 < \\alpha < 2 < \eta \, determine the range of values for the constant \ a \.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.58
'Determine the number of real solutions of the quadratic equation .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.59
'Solve the following system of simultaneous equations.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.60
'Chapter 1\nNumbers and Expressions\n23\nExample\n(1) Find the expression that when summed with 2x^2-3x+1 results in x^2+2x.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.61
'What is the range of the constant a when one real solution of the quadratic equation 2x^{2}-3ax+a+1=0 is in the range 0<x<1 and the other real solution is in the range 4<x<6?'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.63
'What is the range of existence of the roots of a quadratic equation?'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.64
'Find the number of real solutions for the following quadratic equations.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.65
'Given line segments of length a and b, find the positive solution to the quadratic equation x^{2}-a x-b^{2}=0 and draw a line segment with that length.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.66
'Find the range of values for the constant such that the quadratic equation has two distinct real solutions within the range .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.67
'Determine the range of values for the constant when the quadratic equation satisfies the following conditions: (1) Has positive and negative roots. (2) Has two distinct negative roots.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.68
'In all permutations formed by the 8 letters of YOKOHAMA, find the number of permutations that contain at least one of the sequences AO or OA.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.69
'Transform the given mathematical expression into a different form.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.70
'Chapter 2 Sets and Propositions\n(2) Solve the following equation\n\\[(p q+6)+(3 p+q) \\sqrt{2}=8+7 \\sqrt{2}\\]\nwhere p and q are rational numbers.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.71
'How many ways are there for 4 men and 5 women to line up in a row with the following conditions? (1) All 4 men are adjacent (2) Men are not adjacent to each other'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.72
'Please state the converse, contrapositive, and inverse of the proposition.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.73
'Determining coefficients from maximum and minimum values (3)'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.74
'Let PR denote all real numbers. Choose the appropriate term to fill in the square (∎) from below (ア)–(工). (1) x=2 is a ∎ for x^2-5x+6=0. (2) ac=bc is a ∎ for a=b. (3) a=b is a ∎ for a^2+b^2=2ab. (ア) Necessary and sufficient condition (イ) Necessary but not sufficient condition (ウ) Sufficient but not necessary condition (工) Neither necessary nor sufficient condition'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.75
'54 (2), (3); (2) maximum at x=2 is 7, minimum at x=0 is 3; (3) maximum at x=2 is 5, minimum at x=-1.5 is -13'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.76
'What is the range of values for for which the quadratic equations , both have real number solutions?'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.77
'Given x ≥ 0, y ≥ 0, and 2x+y=8, find the maximum and minimum values of xy.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.78
'Use the quadratic formula to solve the following 2nd-degree equations.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.79
'Find the linear function based on the following conditions to maximize profit.\n\n(1) When x is 250, y is 300.\n(2) When x is 300, y is 250.\n(3) Also holds when x = 350, y = 200.\n\nFurthermore, using revenue xy and expenses 120y + 5000, let z be the profit and find the value of x that maximizes z and the maximum profit at that time.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.80
'Determine the truth values of the following propositions.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.82
'If one of the solutions of the quadratic equation is , find the other solution.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.83
'(Example) 1) When 1/x + 1/y + 1/z = 1 (0 < x < y < z), we use 1/z < 1/y < 1/x to get 1 = 1/x + 1/y + 1/z < 3/x, hence x = 1,2.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.84
'Solve the following inequality in terms of x. Where a is a constant. \\[ x^{2}-\\left(a^{2}+a\\right) x+a^{3} \\leqq 0 \\]'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.85
'Solve the following system of simultaneous equations.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.87
'At a certain school, it was decided to completely drain the water from the pool for cleaning. However, it is assumed that a constant amount is drained per minute with a pump. Let the remaining volume of water in the pool after t minutes of drainage be V m³.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.89
'Solve the following system of simultaneous equations.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.90
'At a party with 5 participants, where each person prepares a gift and a drawing is held to distribute all gifts. The number of ways in which two specific people, A and B, receive the gifts they prepared, while the remaining three people receive gifts other than the ones they prepared is denoted by A. The number of ways in which only one person receives the gift they prepared is denoted by B.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.91
'Basic Example 30 Number of Integer Solutions (Using Combinations with Repetition)'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.92
'Find the range of existence of solutions for a quadratic equation with x < 2 and x > 2.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.94
'Solve the inequality for x. Here, a is a constant. \ x^{2}-3 a x+2 a^{2}+a-1>0 \'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.95
"Please provide 'Linear Diophantine Equations' and its page."
A. ...
Ask AI tutor for answer!
Join our DiscordQ.96
'Find the coordinates of the two points of intersection of the two parabolas y=x^2-x+1 and y=-x^2-x+3.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.97
'31 (1) \ x=6,-2 \\n(2) \ x \\leqq-5, \\quad \\frac{1}{5} \\leqq x \'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.98
'Find the largest 4-digit natural number that leaves a remainder of 2 when divided by 11 and a remainder of 5 when divided by 6.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.99
'Solve equations and inequalities involving absolute value: (1) , (2) , (3) '
A. ...
Ask AI tutor for answer!
Join our DiscordQ.00
'Consider the proposition p⇒q (where p is the hypothesis and q is the conclusion). Let P be the set of all elements satisfying condition p, and let Q be the set of all elements satisfying condition q. The proposition p⇒q being true is equivalent to P ⊆ Q. Please determine the truth value of this proposition.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.02
'For the function , find the range of values of that satisfy the following conditions:'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.03
'Let a and b be real numbers, and consider the parabolas represented by the functions y=4 x^{2}-8 x+5 and y=-2(x+a)^{2}+b. Find the values of the constants a and b when the vertices of these parabolas coincide.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.04
'When the quadratic equation has equal roots, find the value of the constant and the equal roots at that time.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.06
'At x=3/2, the minimum value is 3/2, and there is no maximum value.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.07
'Use the formula for solving a quadratic equation ax^{2}+bx+c=0 to solve the equation.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.08
'Find the value of the constant k when the parabola y = x^2 + (2k-3)x-6k cuts a segment of length 5 from the x-axis.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.09
'There is a lottery where three dice are rolled at once. There are multiple venues for the lottery, each with different winning conditions.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.10
'Find the range of values for the constant m when the quadratic equation x^2 + 3x + m - 1 = 0 has no real solutions.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.11
'For the quadratic equation , find the range of values for the constant when it has the following:'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.12
'For example, there are many integer solutions to the equation x+y=10. Find any three integer solutions to this equation.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.14
'When the solutions of the quadratic equation are as follows, find the range of the constant . (1) Having two distinct real solutions. (2) Having real solutions. (3) Having no real solutions.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.15
"Chapter 5 Quadratic Equations and Quadratic Inequalities\nLet h meters be the height above the ground of a ball thrown directly upwards at a certain velocity x seconds after the launch. When the value of h is given by h=-5x^2+40x, at what range of x values is the ball's height between 35 meters above the ground and 65 meters above the ground?"
A. ...
Ask AI tutor for answer!
Join our DiscordQ.16
'Find the values of \ a, b \ such that \ P=4 \ for the point \ x, y \ with values \\( (2, 1) \\).'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.17
'Solve the following equations. 1. Fundamental 86 - Solving quadratic equations using factorization. 2. Fundamental 87 - Solving quadratic equations using the formula for roots. 3. Fundamental 88 - Conditions for having real solutions (1)'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.19
'Determine the range of values for the constant so that the quadratic equation satisfies the following conditions: (1) has two distinct negative roots. (2) has a positive root and a negative root.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.20
'24 (1) Inverse: If at least one of x, y is a negative number, then x+y=-3, False Converse: If x≥0 and y≥0, then x+y≠-3, Contrapositive: If x+y≠-3, then x≥0 and y≥0.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.21
'2 (1) 5 expressions(2)(ア)2 expressions, constant term 2 y^{2} + 5 y - 12(1)2 expressions, constant term 6 x^{2} - 6 x - 12(ら)2 expressions, constant term -12'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.22
'When the roots of the quadratic equation have the following characteristics, find the range of values for the constant .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.23
'The condition for a graph to always be above the x-axis is that the graph is a concave down parabola and does not have any points of intersection with the x-axis. Therefore, the discriminant of the quadratic equation mx^2 + 3x + m = 0 is denoted as D.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.24
'Solve the following equations and inequalities. (1) |(√14-2)x+2|=4 (2) 3|x-1|≤4 (3) x+|3x-2|=3'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.25
'Translate the given text into multiple languages.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.26
'Find the value of the constant and the repeated root of the quadratic equation when it has repeated roots.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.27
'When the quadratic equation has two distinct real roots, find the range of values for the constant .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.28
'For the function f(x) = x^2 - 2ax - a + 6 in mathematics I TR, the range of values of the constant a for which f(x) > 0 for all real numbers x is from A to T. Also, the range of values of a for which f(x) ≥ 0 always holds for -1 ≤ x ≤ 1 is from ウ to エ.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.29
'In two inequalities, first replace > with = and then solve the quadratic equation. To solve the quadratic inequality x^2-6x+3>0, first solve the equation x^2-6x+3=0. Using the formula for roots, x=(-(-3) ± √((-3)^2-1*3))/1=3 ± √6.\n\nThe solution to x^2-6x+3>0 is to find the range of x values on the graph of y=x^2-6x+3 where y>0, which is x<3-√6, 3+√6<x.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.31
'At x=-1, the maximum value of 71(2) is 5, there is no minimum value.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.32
'(1) Solve the equation 2 x^{2}+x-1=0 \\\\\\\n(2) Solve for x=-1, \\ \\frac{1}{2} \\\\\\\n(3) Given \\theta=60^{\\circ}, \\ 180^{\\circ} \\\\\\\n'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.33
'The classification of solutions of a quadratic equation can be determined by examining the sign of the discriminant .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.34
'Find the range of constants a that satisfy the following conditions for the equations x^2+ax+a+3=0 (1) and x^2-2ax+8a=0 (2):'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.35
'Find the intersection point of the two lines 2x + 3y = 7 (1) and 4x + 11y = 19 (2), and the equation of the line passing through the point (5,4).'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.36
'Find the values of when the 3rd-degree function satisfies .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.38
'(1) Find the solutions of the equation 2x^2 - 2√6x + 3 = 0.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.40
'Determine the number of solutions to the equation 4cos²x-2cosx-1=a within the range -π < x ≤ π.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.41
'Find the value of a that satisfies the given equations.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.43
'Find the values of k for which x = -1, -4 and x = 1, 4 are solutions.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.44
'The solutions to the cubic equation are . Find the values of constants , .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.45
'Find the value of a that satisfies the following equation. a=2'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.46
'Regarding the lines l: 2x - y + 3 = 0, m: 3x - 2y - 1 = 0, answer the following questions.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.47
'Determine the range of the constant a so that the two quadratic equations 9x^{2}+6ax+4=0 (1) and x^{2}+2ax+3a=0 (2) satisfy the following conditions. (1) Both have complex roots (2) At least one has complex roots (3) Only (1) has complex roots'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.48
'When a = -1, x = 2, -1\nWhen a = 0, x = 0, 2\nWhen a = 8, x = -4, 2'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.50
'Find the range of values for c so that the equation x^3-6x+c=0 has two distinct positive roots and one negative root.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.51
'Determine the following for the sequence {\ \\left\\{a_{n}\\right\\} \} where the sum of the first \ n \ terms, denoted as {\ S_{n} \}, satisfies the relation {\ S_{n}=-2 a_{n}+4 n \}:\n(1) Find the first term {\ a_{1} \}.\n(2) Find the relation between {\ a_{n} \} and {\ a_{n+1} \}.\n(3) Find the general term of the sequence {\ \\left\\{a_{n}\\right\\} \}.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.52
'Find all real numbers such that the cubic equation has exactly 2 real roots.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.53
'Use the formula for solving the quadratic equation ax^2 + bx + c = 0 to find the solutions.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.54
'After reading the conversation between Hanako and Taro about the problem, answer the following question.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.55
'Determine the values of constants a and b that satisfy the following conditions:\n(1) When x-1 is a factor of x^3-3x^2+a, the remainder is 2.\n(2) When 2x+1 is a factor of 2x^3-3x^2+ax+6, it divides evenly.\n(3) When x+2 divides x^3+ax^2-5x+b, the remainder is 8, and when x+1 is a factor.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.56
'Transform the given text into multiple languages.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.57
'Determine the value of the real number k so that the equation (1 + i)x^{2} + (k + i)x + 3 + 3ki = 0 has real solutions. Find these real solutions.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.58
'The two numbers that sum to -2 and multiply to 3 are the solutions to the quadratic equation x^2 + 2x + 3 = 0.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.59
'Find the value of the constant a such that the equation 2x^3 - (3a + 1)x^2 + 2ax + 4 has two distinct real roots.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.61
'Chapter 2\nComplex Numbers and Equations\nShow that for the two solutions α, β of the quadratic equation 2x^2 + 4x + 3 = 0, the following holds:\n1. (α-1)(β-1)=9/2\n2. (α-1)^3 + (β-1)^3 = -10'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.62
'If the quadratic equation has real roots, then and the solutions are .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.63
'Create a quadratic equation with the sum as \ p \ and the product as \ q \.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.64
'Find the values of real numbers x, y that satisfy the following system of equations.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.65
'For the quadratic equation x^2 - 2ax + a + 6 = 0, find the range of values for the constant a that satisfy the following conditions: (1) Having one solution greater than 1 and one solution less than 1. (2) All solutions are greater than or equal to 2.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.66
'Create a quadratic equation with the roots α and β.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.67
'Find the values of the constants p, q, and r such that the equation p x^2 + q y^2 + r z^2 = 12 holds for all real numbers x, y, z satisfying the system of equations 2x + y - 3z = 3 and 3x + 2y - z = 2.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.68
'When the equation has roots at , find the values of constants . Also, find the other roots in this case.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.69
'The sum and product of the two roots of the quadratic equation are the roots of the quadratic equation . Determine the values of constants .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.70
'The profit per day is (a x + 3 y) 10,000 yen. If we let ax + 3 y = l ....5), then (5) represents a straight line with a slope of -a/3 and a y-intercept of l/3.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.71
"Let's consider the maximum and minimum values of -3x+y under the same conditions as A (let's call this C). If we express -3x+y=k, then y=3x+k。"
A. ...
Ask AI tutor for answer!
Join our DiscordQ.72
'Let A be a polynomial. When x^6 - 6x^3 + 5x^2 - 4x + 10 is divided by A, the quotient is A and the remainder is 5x^2 - 4x + 1. Find the polynomial A.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.73
'Determine the values of constants a, b, c, and d so that the following equations hold true for x:'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.74
'Find the range of values for the constant p such that the equation x^3-3p^2x+8p=0 has three distinct real solutions.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.75
'For the quadratic equation in terms of x, 8x^2-4x-a=0, with two solutions being sin θ and cos θ, find the value of the constant a and the 2 solutions of the equation. [Similar to Keio University] From the relationship between the solutions of the quadratic equation and the coefficients, sin θ+cos θ=-\\frac{-4}{8}=\\frac{1}{2}, sin θ cos θ=-\\frac{a}{8}. Squaring both sides of (1) gives sin^2 θ+2sin θ cos θ+cos^2 θ=\\frac{1}{4}, hence 1+2sin θ cos θ=\\frac{1}{4}, which implies sin θ cos θ=-\\frac{3}{8}. Substituting this into (2) gives -\\frac{a}{8}=-\\frac{3}{8}, thus a=3. Therefore, the given quadratic equation is 8x^2-4x-3=0. Solving this equation, the two solutions are x=(1±√7)/4.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.76
'Basic Column Question 18 Determining the Coefficients of an Identity'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.77
'Use recurrence relation to find the general term of the sequence.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.78
'Given the quadratic equation with two roots , find the value of the following expressions.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.79
'The cubic curve y = ax^3 + bx^2 + cx + d touches the x-axis at x = 2 and the equation of the tangent line at the origin is y = -2x. Find the values of the constants a, b, c, and d.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.80
'When (x + y) / 2 = (y + z) / 5 = (z + x) / 7 (not equal to 0), find the value of (xy + yz + zx) / (x^2 + y^2 + z^2).'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.81
'The quartic equation x^4+ax^3+7x^2+bx+26=0 has two common roots with the quadratic equation x^2+2x+2=0 and the sum of the common roots is 37. [Tokushima Bunri University] (1) Find the values of the real constants a, b. (2) Find the remaining roots of the quartic equation.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.82
'Since this line passes through the point (2, -4), -4 = 2(a-1) • 2 - a^2. Simplifying, we get a^2 - 4a = 0. Therefore, a(a-4) = 0, which means a = 0, 4. Hence, the equation of the tangent line we seek is y = -2x when a = 0, and y = 6x - 16 when a = 4.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.83
'Find the range of possible values for x given that x, y, z are real numbers satisfying x+y+z=0 and x^2-x-1=yz. Determine the maximum and minimum values of x^3+y^3+z^3 and the corresponding values of x.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.84
'(3) For a geometric sequence where the first term a and common ratio r are both real numbers, if the sum from the first term to the nth term is Sn, and when Sn=3 and Sn=27. Find the values of a, r.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.85
'In the mathematics \ \\mathbb{I} \ EX\\nLet the two solutions of the quadratic equation \\( 2 x^{2}-2(2 a-1) x-a=0 \\) be \ \\sin \\theta, \\cos \\theta \. Find the positive constant \ a \ and the values of \ \\sin \\theta, \\cos \\theta \. Given that \ 0 \\leqq \\theta \\leqq \\pi \.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.87
'Let a be a constant. Prove that at least one of the two quadratic equations x^{2}+2ax+3a=0 and 3x^{2}-2(a-3)x+(a-3)=0 has real roots.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.88
'Regardless of the constant a, the circle C1 passes through the fixed point A. Find the coordinates of this fixed point A.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.89
'Example 76: Equation of a line passing through a fixed point'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.91
'Determine the types of solutions for the following quadratic equations:'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.92
'Solve the system of equations x^2-3xy+y^2=19, x+y=2.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.93
'When the equation represents a circle, find the range of values for the constant .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.94
'For different numbers a, b, if the sequence √3, a, b is an arithmetic sequence, and the sequence a, √3, b is a geometric sequence, then a = square root of 3, and the common ratio of the geometric sequence is -3.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.96
'Find the solutions of the following equations. (1) (2) '
A. ...
Ask AI tutor for answer!
Join our DiscordQ.97
'The coordinates of the intersection points are determined by the solutions of the following 3 sets of simultaneous equations: (1) y=x²-4 and y=x-2 (2) y=x²-4 and y=-1/2 x-7/2 (3) y=x-2 and y=-1/2 x-7/2. Find the intersection points in (3) and calculate the area S of that region.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.98
'Determine the types of solutions for the following quadratic equations. Here, a is a constant.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.99
'When having a repeated root (i.e. α=β), because α=β, then α=β, so a_{n+2}-αa_{n+1}=α(a_{n+1}-αa_{n})'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.00
'Translate the given text into multiple languages.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.01
'If one of the solutions of the cubic equation x^3 + ax^2 + bx + 10 = 0 is x = 2 + i, find the values of the real constants a, b and the other solution.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.03
'Let the three solutions of the cubic equation be α, β, and γ. Then, create another cubic equation with solutions α+β, β+γ, and γ+α. Assume the coefficient of x^3 to be 1.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.04
'In mathematics, the part where y>0 of the curve y=\\frac{x^{2}}{4} and the line y=2 x+a have two distinct points of intersection in the range x>0.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.05
'For the sequence defined by {an}, where a1=-1 and an+1 = an^2 + 2nan - 2 (n=1,2,3,...), infer the general term an and prove its correctness using mathematical induction.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.07
'When dividing a polynomial P(x) by x-2, the remainder is 3, and by x+3, the remainder is -7. Find the remainder when P(x) is divided by (x-2)(x+3).'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.09
"When this line passes through point A(0, a), in the graph of the cubic function, different tangents correspond to different points of tangency. Therefore, when the equation of t has three distinct real solutions, three tangents can be drawn from point A to the curve. Now, if we define h(t)=2t^3-9t^2+7+a, then h'(t)=6t^2-18t=6t(t-3). The table for the increase and decrease of h(t) is as follows."
A. ...
Ask AI tutor for answer!
Join our DiscordQ.10
'Solve the system of equations {\\\left\\{\egin{\overlineray}{l}\\cos x-\\sin y=1 \\\\ \\cos y+\\sin x=-\\sqrt{3}\\end{\overlineray}\\right.\}. Given that {\0 \\leqq x \\leqq 2 \\pi, 0 \\leqq y \\leqq 2 \\pi\}.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.11
'Choose one that fits in the blank from (0) to 3.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.12
'Let α and β be the two roots of the quadratic equation x^2 + 3x + 4 = 0, then create a quadratic equation with roots α^2 and β^2'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.13
'When only producing product Q (without producing product P), at (x, y) = (0,52). The profit ax + 3y reaches its maximum only when the slope equals -5/3. It reaches the maximum value only when -5/3 < -a / 3 < 0 and -5/3.5/3.5/3 takes place. When -5/3 < -a / 3 < 0, the maximum value of profit is 156 million yen (a = 0, y = 52).'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.14
'A line passing through point A(-1,0) with slope a is denoted by l. The parabola y=1/2*x^2 intersects the line l at two different points P and Q. (1) Find the range of values for the slope a. (2) Express the coordinates of the midpoint R of line segment PQ in terms of a. (3) Plot the locus of point R on the xy plane.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.15
'When the cubic equation has three real solutions , answer the following questions.\n(1) Find the range of values for the constant .\n(2) Find the range of values for .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.16
'When the polynomial P(x) is divided by x-2, the remainder is 13, and when divided by (x+1)(x+2), the remainder is -10x-3. In this case, find the remainders when P(x) is divided by (x+1)(x-2)(x+2) and (x-2)(x+2) respectively.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.17
'Let k be a constant. Determine the number of distinct real solutions of the cubic equation .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.18
'Determine the values of the constants a and b that satisfy the following conditions.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.19
'Prove that at least one of the two quadratic equations has real roots: , .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.20
'The profit maximizes only when the slope of the line (ax + 3y) at (50, 20) satisfies -5/2 < -a/3 < -3/4. Therefore, a should lie between 9/4 and 15/2.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.21
'Let the two roots of the quadratic equation be , then form a quadratic equation with solutions as the two numbers .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.22
'Let a and b be constants. The function f(x) = x^3 - 9x^2 + ax + b has a local extremum at x = 1, and the equation f(x) = 0 has one positive and one negative real root. Find the values of a and b.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.23
'At time 0, two particles are located at vertex A of triangle ABC. These particles move independently, with each moving to an adjacent vertex with equal probability every 1 second. Let n be a natural number, and let pn be the probability that these two particles are at the same point after n seconds from time 0. (1) Find pn. (2) Express pn+1 in terms of pn. (3) Express pn in terms of n. [Similar to Kyoto University]'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.24
'Find the general term of the sequence {an} determined by the following conditions.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.25
'Find the values of constants a and b. When one of the solutions of the quadratic equation x^2+ax+b=0 is x=2-3i, find the values of a and b. Also, find the other solution of this equation.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.27
'Let x, y, z be real numbers satisfying y+z=1 and x^2+y^2+z^2=1.\n(1) Express yz in terms of x. Also, find the range of values for x.\n(2) Express x^3+y^3+z^3 as a function of x, and determine the maximum and minimum values, as well as the corresponding x value.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.28
'When (x, y) ≠ (50, 20), the profit does not reach its maximum. Consider the condition. Pay attention to the slopes of lines (2), (3), (5).'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.29
'Find the equation of the line passing through the point (2,1) with a slope of −\\frac{1}{2}.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.30
'Solving Triangular Inequalities and Equations (Quadratic Equations)'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.32
'Find all the values of that satisfy the equation for all values of .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.33
'To ensure that the equation has real solutions, determine the value of the real number .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.34
'Solve the following equations and inequalities for 0 ≤ θ < 2π:'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.35
'When the two solutions are represented by sinθ and cosθ respectively, find the value of k and determine the two solutions.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.37
'Find the values of real numbers x and y that satisfy the equation (2+i)x - (1-3i)y + (5+6i) = 0.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.38
'Find the first term and common difference of an arithmetic sequence where the sum of the first 10 terms is 100 and the sum of the first 20 terms is 350. Also, find the sum of terms 21 to 30 in this sequence.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.39
'Find the positive constant a and the values of sin θ, cos θ that are the two solutions of the quadratic equation 2x^2 - 2(2a-1)x - a = 0. Here, 0 ≤ θ ≤ π.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.41
'Let there be a sequence of positive numbers a_{1}, a_{2}, \\cdots \\cdots, a_{n}, \\cdots \\cdots, satisfying the recurrence relation \\[\\sqrt{2} a_{n}{ }^{5}=a_{n-1}^{6}(n=2,3,4, \\cdots \\cdots)\\]. In this case, express a_{n} in terms of a_{1} and n.[Central University]'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.42
'Let the quadratic equation 2x^{2} + 4x + 3 = 0 have two solutions α and β. Find the following values:\n(1) (α-1)(β-1)\n(2) (α-1)^{3} + (β-1)^{3}\n[Similar to Keio University]'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.43
'Two particles are located at vertex A of triangle ABC at time 0. These particles move independently, with each moving to an adjacent vertex with equal probability every second. Let n be a natural number, and let pn be the probability that these two particles are at the same point after n seconds from time 0.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.45
'Find the sum and product of the two solutions of the following quadratic equations:'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.46
'Basic example 632 with a condition of repeated roots\nDetermine the value of the real constant a such that the 3rd degree equation x^{3}+(a-1)x^{2}+(4-a)x-4=0 has a double root.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.48
'Let EX be 0 and a non-zero constant, and i be the imaginary unit. Suppose there exists a real number x=α that satisfies the equation x^{2}+(3+2 i) x+k(2+i)^{2}=0, find (1) the values of k and α. (2) Find all complex numbers that satisfy this equation.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.49
'Find the general term of the following sequences'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.50
'The equation x^4+ax^3+7x^2+bx+26=0 has two common solutions, one of which is a solution to the quadratic equation x^2+2x+2=0.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.51
"Dividing the polynomial P(x) by x-2 results in a remainder of 13, and dividing by (x+1)(x+2) results in a remainder of -10x-3. Let's divide P(x) by (x+1)(x-2)(x+2), with the quotient being Q_1(x) and the remainder being ax^2+bx+c. Then, the following equation holds: P(x)=(x+1)(x-2)(x+2)Q_1(x)+ax^2+bx+c. Since the remainder of P(x) divided by x-2 is 13, we have P(2)=13....(2). Furthermore, dividing P(x) by (x+1)(x+2) gives the quotient Q_2(x) and a remainder of -10x-3."
A. ...
Ask AI tutor for answer!
Join our DiscordQ.52
'If the two solutions of the quadratic equation ax^2+bx+c=0 are α and β, then what are α+β and αβ equal to?'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.53
'Find the general term of the sequence {an} determined by the following conditions.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.54
'Find the values of θ that satisfy the equation cos 3θ - cos 2θ + cos θ = 0 for 0 ≤ θ < 2π.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.55
'Find the conditions for PR simultaneous equations 3x - 2y + 4 = 0, ax + 3y + c = 0 to have:\n(1) a unique solution\n(2) no solution\n(3) infinitely many solutions'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.57
'When the equation has one positive solution and one negative solution each, find the range of possible values for the constant .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.58
'Find the values of constants such that holds for all satisfying .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.59
'Using the relationship between the solutions of a cubic equation and its coefficients to solve a cubic equation'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.62
"Let's consider the maximum and minimum values of x+3y under the same conditions as A (denoted as B). Considering x+3y=k, y=-(1/3)x+k/3."
A. ...
Ask AI tutor for answer!
Join our DiscordQ.63
'Let α and β be the two roots of the quadratic equation x^{2}-x+8=0. Find the values of the following expressions. [Similar to Hannan]\n(1) α^{2}+β^{2}\n(2) α^{4}+β^{4}\n(3) \\frac{\eta}{1+α^{2}}+\\frac{\\α}{1+β^{2}}'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.64
'Solve the following equations: \n\\n\egin{\overlineray}{l}\n\\alpha^{2}+\\sqrt{3} \eta=\\sqrt{6} \\\\\n\eta^{2}+\\sqrt{3} \\alpha=\\sqrt{6}\n\\end{\overlineray}\n\'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.66
'Since the equation is , the equation becomes , hence , that is . Therefore, .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.67
'Find the number of real solutions of the equation . Also, if there is only one solution, find that solution.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.68
'When 2 x+\\frac{1}{2 x}=\\sqrt{7}, find the values of the following expressions. (1) 4 x^{2}+\\frac{1}{4 x^{2}} (2) 8 x^{3}+\\frac{1}{8 x^{3}} (3) 64 x^{6}+\\frac{1}{64 x^{6}}'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.69
'Assuming the equation has two positive equal roots, find the value of the constant .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.71
'When two quadratic equations have a unique real number as a common solution, find the value of the real constant and the common solution at that time.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.72
'Determine the range of values for the constant such that the quadratic equation has two distinct real roots within the range .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.74
'Consider the following problem: For a>1, is a+1/2>3/2 true? If true, then what is the minimum value of f(a+1)=-2(a+1)^2+6(a+1)+1=-2a^2+2a+5?'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.75
'When k=0, the common solution is x=0; when k=\x0crac{5}{22}, the common solution is x=-\x0crac{1}{2}'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.76
'(3) \ \\left\\{\egin{\overlineray}{l}2 x+4>x^{2} \\\\ x^{2}>x+2\\end{\overlineray}\\right. \'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.77
'Find the length of the line segment that the graph of the quadratic function y=-2x^{2}-3x+3 cuts from the x-axis.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.78
'Let k be a constant. Determine the number of distinct real solutions of the equation |x²+2x-3|+2x+k=0.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.79
'One of the solutions is x=0, so f(0)=0 implies -a+1=0. Therefore, a=1. In this case, the equation becomes x^{2}+3x=0, which leads to x(x+3)=0. Hence, the solutions are x=-3,0, but they do not satisfy the condition. What is the range of values for a that we are looking for?'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.80
'Determine the value of the constant k such that the equations 2 x^{2}+k x+4=0 and x^{2}+x+k=0 have exactly one common real solution, and find this common solution.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.81
'(1) If the solutions of the quadratic equation are 2 and -4, find the values of the constants .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.82
'Let α be one of the solutions of the quadratic equation x^2 + 4x - 1 = 0, then α-1/α= A, and α^3-1/α^3= B.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.83
'Assume a, b, c are all positive. Starting from a>0, the graph of y=a x^{2}+b x+c is a concave parabola, for x<p and for sufficiently large |x|, a x^{2}+b x+c>0 holds. Similarly, for x<p and for sufficiently large |x|, b x^{2}+c x+a>0, and c x^{2}+a x+b>0. Therefore, there exist negative numbers x with sufficiently large absolute values that are in I but not in J, which contradicts I=J. Hence, at least one of a, b, c is less than or equal to 0. Combining this with (1), at least one of a, b, c is zero.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.85
'From [1], [2], when the equation |(x-2)(x-4)|=a x-5 a+\x0crac{1}{2} has 4 distinct real roots, what is the range of values for a?'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.86
'Let a be a constant. Find the number of real solutions to the equation (a-3)x^2 + 2(a+3)x + a+5 = 0. Also, if there is one solution, find that solution.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.87
'Find the range of values for the constant such that the equation has at least one real solution in the range .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.88
'When one of the solutions of the quadratic equation is , find the value of the constant and the other solution.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.90
'68 (1) x = -1, 1/3 (2) x = -4 ± √6/3 (3) x = 1 - √3 (4) x = ±1, ±3/2'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.91
'Translate the given text into multiple languages.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.92
'Simplify to get 4 sin^2(θ) - (2+2√2) sin(θ) + √2 < 0 Let sin(θ) = t, then when 0° ≤ θ ≤ 180°, 0 ≤ t ≤ 1. The inequality becomes 4t^2 - (2 + 2√2)t + √2 < 0, hence (2t - 1)(2t - √2) < 0, so 1/2 < t < √2/2 (1). The common range is 1/2 < t < √2/2, so, 1/2 < sin(θ) < √2/2. Solving this yields 30° < θ < 45°, 135° < θ < 150°.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.93
'Let EX 90 be a constant. Find the range of values of a when the equation |(x-2)(x-4)|=a x-5 a+1/2 has four distinct real number solutions.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.95
'Let k be a constant. Determine the number of distinct real number solutions of the equation |x²-x-2|=2x+k.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.96
'Find the quadratic function that satisfies the following conditions: (2) intersects the x-axis at points (-1,0) and (2,0), and passes through the point (3,12).'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.97
'Let a be a constant. For -1 ≤ x ≤ 1, consider the function f(x)=x2+2(a−1)x and answer the following question: (1) Find the minimum value.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.98
'Find the range of values for the constant such that the equation has at least one real solution in the interval .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.00
'For the two inequalities and , answer the following 9 questions. Where is a constant with .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.01
'Find the number of real solutions to the equations.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.03
'Find the general solution of the quadratic equation by equivalent transformations without using α.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.04
'Please indicate the first occurrence page of the following terms. ①General form (quadratic equation) ②Double root ③Gauss symbol'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.06
'Determine the value of the constant k such that the equations x^{2}-(k-3)x+5k=0 and x^{2}+(k-2)x-5k=0 have only one common solution, and find that common solution.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.07
'When one of the solutions of the equation is , find the value of the constant and the other solution.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.08
'Prove that there do not exist 5 distinct real numbers that simultaneously satisfy both propositions (A) and (B):\n(A) Among the 5 numbers, choosing any one of them will result in the sum of the remaining 4 numbers being smaller than the chosen number.\n(B) Choose any 2 out of the 5 numbers. The larger of the two numbers is greater than twice the smaller number.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.09
'When 33\na>3, x>−\\frac{b}{a−3}, when a=3 and b>0, the solution is all numbers, when a=3 and b=0, there is no solution, when a<3, x < −\\frac{b}{a−3}'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.10
'There is at least one natural number n such that n²-5n-6=0'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.11
'Question 2 Find the quadratic function passing through the intersection points of a parabola and a line. Let f(x)=x-1, g(x)=-x^2+5x-2, and let the line y=f(x) and the parabola y=g(x) have two common points A, B. Also, consider point P(2,-5).'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.12
'When p≤0≤q, f(x) takes the minimum value -1 at x=0, and the minimum value is p, so p=-1, which satisfies p≤0.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.13
'Solve the following equations and inequalities: (1) |x-3|+|2x-3|=9 (2) ||x-2|-4|=3x (3) |2x-3| ≤ |3x+2| (4) 2|x+2|+|x-4|<15'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.14
'Problem of finding common solution: Find the common solution of the equations f(x)=0 and g(x)=0. Explained based on example 102.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.15
'Translate the given text into multiple languages.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.16
'If one of the solutions of the quadratic equation x^2 + (a+4)x + a - 3 = 0 is a, find the other solution.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.18
'(1) When a = 4, what is the value of x? (2) What is the range when a > 7?'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.19
'Determine the range of values for the constant so that the quadratic equation has real solutions.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.20
'When a≠0, the discriminant of the quadratic equation f(x)=0 is denoted by D.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.21
'Solve the quadratic equation using the formula for finding roots, and determine the coordinates of the points of intersection with the -axis.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.22
'(3) When the equation has two solutions , find the values of constants .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.23
'Consider the function f(x)=x^{2}-2 a x+a^{2}+2 a-3, where 0 ≤ x ≤ 1.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.24
'Determine the number of real solutions for the quadratic equation .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.25
'Let the two distinct real solutions of the quadratic equation be , such that . Determine the range of values for the constant .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.27
'Consider the quadratic equation . Find the following:\n(1) Range of values for to have two different solutions greater than 1\n(2) Range of values for to have two different solutions less than 1\n(3) Range of values for to have one solution greater than 1 and one solution less than 1'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.28
'Find the value of the constant m and the repeated root when the quadratic equation x^2 + (m-8)x + m = 0 has a repeated root. (Nagoya Shoka University)'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.29
'Find the values of x that satisfy x^2 - 2(x+1) = 2, where x + 1 ≥ 0, meaning x ≥ -1.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.30
'When the equation (m+1)x² + 2(m-1)x + 2m-5 = 0 has exactly one real solution, find the value of the constant m.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.32
'Consider the quadratic equation with two distinct real roots . Determine the range of values of the constant such that .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.33
'(1) When one of the solutions to the quadratic equation is -2, find the value of the constant .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.34
'Find the range of values for the constant when the quadratic equation satisfies the following conditions:\n(1) It has two distinct roots, both greater than 2.\n(2) It has one root greater than 2 and one root smaller than 2.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.35
'Determine the value of the constant a so that the two graphs have only one point in common.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.37
'Determine the values of the constants a and b such that the solution of the quadratic inequality in x, ax^2+9x+2b>0, is 4<x<5.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.38
'Solve (1) x-1=0 or x+2=0. (2) x=0 or x+1=0. (3) Find x= ±√(8/9).'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.39
'When a=-2, there is no solution; when a=2, the solution is all real numbers; when a≠±2, x=-1/(a+2). (1) When p=-1, x=1/2; p=1, x=-1/2; when p≠±1, x=-1/(p+1), -1/(p-1).'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.40
"Let's give it a try. \\\qquad\\nQuestion 1 Let \ a \ be a real number. Choose the correct option that fits in the following \\\square\ from the options below (1)–(4).\n(1) \ |a+1|=2 \ for \ a^{2}+2 a-3=0 \ means \\\square\ .\n(2) \ |a-1|<2 \ for \ a^{2}-1<0 \ means \\\square\ .\n(3) \ 1<|a|<2 \ for \ -1<a<2 \ means \\\square\ .\n(1) Necessary but not sufficient condition\n(2) Sufficient but not necessary condition\n(3) Necessary and sufficient condition\n(4) Neither necessary nor sufficient condition"
A. ...
Ask AI tutor for answer!
Join our DiscordQ.41
'Find the number of real solutions for x in the quadratic equation x^{2}+(2 k-1) x-3 k^{2}+9 k-2=0.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.42
'Solve the following 2nd degree equations. (1) \ x^{2}-3 x+2=0 \ (2) \ 2 x^{2}-3 x-35=0 \ (3) \ 12 x^{2}+16 x-3=0 \ (4) \ 14 x^{2}-19 x-3=0 \ (5) \ 5 x^{2}-3=0 \ (6) \\( (2 x+1)^{2}-9=0 \\)'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.43
'For f(x) = x² - 2ax - a + 6, find the range of values for the constant a such that f(x) ≥ 0 for all -1 ≤ x ≤ 1.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.44
'(3) Let a = 1, changing the values of coefficients b, c, the equation f(x) = 0 has roots x = 1,3. In this case, b = cosine, c = □ji. Next, fixing the values of coefficients b, c to b = □sa, c = □shi, when decreasing the value of a, consider the roots of the equation f(x) = 0. When 0 < a < 1, the equation f(x) = 0 has root. When a = 0, the equation f(x) = 0 is. When a < 0, the equation f(x) = 0 is. Answer the numbers corresponding to cosine, sa shi. Also, choose the most appropriate option from the following numbers (0-5): (0) has no real number solution, (1) has only one real number solution, which is positive, (2) has only one real number solution, which is negative, (3) has two different positive solutions, (4) has two different negative solutions, (5) has one positive solution and one negative solution'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.45
'Conditions for the existence range of solutions of quadratic equations'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.47
'Find the range of values for the constant a, so that the quadratic equation has two distinct real solutions within the interval .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.48
'If one of the solutions of the quadratic equation x^{2} + (a+4)x + a - 3 = 0 is a, find the other solution.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.49
'(4) \ \\left\\ulcorner x^{2}+y^{2}=1 \\Longrightarrow x+y=0 」\\right. \ is false.\n(Counterexample) \: x=0, y=1\\nAlso, \ \\left\\ulcorner x+y=0 \\Longrightarrow x^{2}+y^{2}=1 」\\right. \ is false as well.\n(Counterexample) \: x=0, y=0\\nTherefore, it is neither a necessary condition nor a sufficient condition.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.50
'Practice 81 Determine the value of the constant k so that the equations x^2 - (k-3)x + 5k = 0, x^2 + (k-2)x - 5k = 0 have a unique common solution, and find that common solution.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.51
'(2) Find the value of x when a = 1 (also find the value of x when a = -3)'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.52
'Find the range of values for the constant a such that the quadratic equation x^2+ax-a^2+a-1=0 has two distinct real solutions within the interval -3<x<3.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.53
'When \ x+y+z=2\\sqrt{3}, xy+yz+zx=-3, xyz=-6\\sqrt{3} \, find the values of \ x^{2}+y^{2}+z^{2} \ and \ x^{3}+y^{3}+z^{3} \ respectively.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.54
'The range of values for where the equations and both have real solutions is A , and the range of values for where at least one of the equations has real solutions is B .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.55
'State the condition for the quadratic equation to have real roots.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.56
'Answer whether (3) is correct or incorrect. If incorrect, indicate which transformation among (A)→(1), (1)→(2), (2)→(3) is incorrect.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.58
'For the two quadratic equations \ x^{2}-x+a=0, x^{2}+2 a x-3 a+4=0 \, find the range of values for the constant \ a \ that satisfy the following conditions:'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.59
'Find the range of values for the constant such that the quadratic equation satisfies the following conditions: (1) It has one positive and one negative root. (2) It has two distinct negative roots.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.60
'Solve the following system of equations:\n(1) x^2 - xy - 2y^2 = 0\n(2) x^2 + xy - y = 1'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.61
'Solve the following system of equations. 2a-b+c=8, a-2b-3c=-5, 3a+3b+2c=9'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.62
'What is the value of m when the two quadratic equations x^2 - (m + 1)x - m^2 = 0 and x^2 - 2mx - m = 0 have exactly one common solution? What is the common solution at that time?'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.63
'Determine the sign (positive, zero, negative) of the following values in the graph of the quadratic function y=ax^2+bx+c as shown on the right:\n(1) a\n(2) b\n(3) c\n(4) b^{2}-4ac\n(5) a+b+c\n(6) a-b+c'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.65
'Determine the range of constants such that the quadratic equation has two distinct real roots within the range .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.66
'Find the values of m such that the two quadratic equations have a common solution, and determine that common solution.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.67
'In the equation x^{2}-2 x-8=0, one of the two real solutions is a solution to the equation x^{2}-4 a x+a^{2}+12=0. Find the value of a.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.68
'For a quadratic equation with one real root in the range and the other real root in the range . Determine the range of the constant .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.69
'Quadratic trigonometric equations and inequalities'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.70
'Find the number of real solutions to the quadratic equation .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.71
'Among the two real solutions of the equation , the smaller solution will be one of the solutions of the equation in terms of . Find the value of in this case.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.72
'Find the range of values for the constant when the quadratic equation has the following types of solutions:\n(1) Two different positive solutions\n(2) One positive solution and one negative solution'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.73
'Let the two distinct real roots of the quadratic equation be , and satisfy . Determine the range of values of the constant .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.74
'When the equation has two solutions , find the values of constants and .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.75
"In Tarou's class, they are planning to set up an okonomiyaki food stall at the cultural festival and are considering the price per piece of okonomiyaki. The following table summarizes past sales data."
A. ...
Ask AI tutor for answer!
Join our DiscordQ.76
'(2) Find the general solution when k = 0. Find the general solution when k = 5 / 22.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.77
'Find the value of x that satisfies the equation `x^2 + 2(x+1) = 2` when `x < -1`.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.78
'For (1) x + 3 y = k, the minimum value of x^2 + y^2 is 4. Find the value of the constant k.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.79
'(1) can be expressed using "and" and "or" as: (x + 5) and (3y - 1) = 0; its negation can be expressed as: not (x + 5) and not (3y - 1) = 0. (2) can be expressed using "and" and "or" as: (x - 2)² + (y + 7)² = 0; its negation can be expressed as: not (x - 2)² and not (y + 7)² = 0.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.80
'For the variable x, there are the following three quadratic equations: x^{2}+a x+a+3=0, x^{2}-2(a-2) x+a=0, x^{2}+4 x+a^{2}-a-2=0. Find the range of values of a for which these quadratic equations have no real solutions.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.81
'Determine the value of the constant ④ as 81k, so that the equations x^2-(k-3)x+5k=0, x^2+(k-2)x-5k=0 have only one common solution in the equation PR, and find the common solution.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.82
'Solve the quadratic equation 5x^2-4=0 and find the coordinates of its intersection points with the x-axis.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.83
'(2) Find the values of constants when the two quadratic equations both have as a root.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.85
'\ \\mathrm{AQ}=\\mathrm{BQ} \ therefore \ \\quad \\mathrm{AQ}^{2}=\\mathrm{BQ}^{2} \\nHence\n```\n\egin{array}{l}\n(x-1)^{2}+(0-2)^{2}+(z-3)^{2}=(x-3)^{2}+(0-2)^{2}+\\{z-(-1)\\}^{2} \\\\\n\\text { Therefore } \\quad-2x-6z+14=-6x+2z+14\n\\end{array}\n```\nSimplifying we get \ \\quad x-2z=0 \\n\ \\mathrm{AQ}=\\mathrm{CQ} \ therefore \ \\quad \\mathrm{AQ}^{2}=\\mathrm{CQ}^{2} \\nHence\n```\n\egin{array}{l}\n(x-1)^{2}+(0-2)^{2}+(z-3)^{2}=\\{x-(-1)\\}^{2}+(0-1)^{2}+(z-2)^{2} \\\\\n\\text { Therefore } \\quad-2x-6z+14=2x-4z+6\n\\end{array}\n```\nSimplifying we get \ \\quad 2x+z=4 \ \ \\qquad \\nBy solving (1), (2) we get \ x=\\frac{8}{5}, z=\\frac{4}{5} \\nTherefore, the coordinates of the required point are \\( \\left(\\frac{8}{5}, 0, \\frac{4}{5}\\right) \\)\n\ \\Leftrightarrow \\mathrm{AQ}=\\mathrm{BQ}=\\mathrm{CQ} \ therefore\n\ \\mathrm{AQ}=\\mathrm{BQ}, \\quad \\mathrm{AQ}=\\mathrm{CQ} \'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.86
'What shape is represented by the set of points z that satisfies the following equations?(1) |z|^2 = 2i(z-\x08ar{z}) (2) 3|z| = |z-4i|'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.87
'Find the conditions for which the equation \\( \\sqrt{x-1}-1=k(x-k)(k<0) \\) has no real solutions.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.88
'For a polynomial , if holds, then prove that is divisible by .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.89
'Summation of infinite series using a recurrence relation'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.90
'From the first statement, x = (t-1/2)^2 + 3/4, hence x ≥ 3/4; also, from the second statement, y takes all real values. These conditions coincide with the range of values that the point (x, y) determined by x = y^2 + y + 1 can take.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.91
'(1) \\left\\{\egin{array}{l}x=t \\\\ y=2 t+2\\end{array}\\right. \\) (2) 2 x-y-8=0'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.93
'(1) When you square both sides of sqrt{2 x-1}=1-x\n \\2 x-1=(1-x)^{2} we get x^{2}-4 x+2=0\\Solving this, we get x=2 ± sqrt{2}\\The x that satisfies (1) is x=2-sqrt{2}\n (2) When you square both sides of |x-3|= sqrt{5 x+9} we get (x-3)^{2}=5 x+9 which simplifies to x^{2}-11 x=0\\Solving this, we get x=0,11\\Both x=0,11 satisfy (1), so these are the solutions'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.94
'Practice: (1) Explain the range of movement of point P from the equation s+t=0 leading to t=-s.\n(2) Analyze the movement range of point P by assuming s+t=k(0 ≤ k ≤ 1).'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.97
'Please prove that the solution to the following quartic equation is x = \\alpha: \a x^{4}+b x^{2}+c=0\Provide an explanation, for example in the form \ a x = \\alpha^{4}+b \\alpha^{2}+c = \\overline{0}\. Also, demonstrate that \x=\\overline{\\alpha}\ is also a solution to the given equation.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.98
'According to Hamilton-Cayley theorem, for matrix A, the following equation holds: A^2-(a+d)A+(ad-bc)E=0. Consider the following cases: [1] When a+d=2, find (a+d, ad-bc) = (2,-8). [2] When a+d ≠ 2, assume A = kE and find the value of k.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.00
'Solve the equation z^{n} = 1 for primitive n-th roots of unity.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.01
'(2) When the equation is transformed, \\ \n\\( \\ \nLet , then , so .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.03
'For f(x)=x^{2}-x-2, find the values of x that satisfy f(f(x))-x=0.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.07
'(3) This holds for all real x. Substituting x=0 into (3) gives -5a-b+15=0 Substituting x=\\frac{\\pi}{2} into (3) gives 5 e^{\\frac{\\pi}{2}}(a-2)=0 Hence, e^{x}\\{-3(a-b+3) \\sin 2 x+6(2-a) \\cos 2 x +(2-a) \\sin x+(a-b+3) \\cos x\\}=0'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.08
"(4) \\ frac { \\ left (1 + y ^ {'2'} \\ right) ^ {3}} {y ^ {'2'} \\ prime \\ prime} = 1"
A. ...
Ask AI tutor for answer!
Join our DiscordQ.09
'Why are entrance exam questions in mathematics often based on specific mathematical phenomena? Please provide three reasons.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.10
'Solve the system of equations and find the number of intersection points.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.11
'Find the solutions to the following equation: 3x^2 - 6x + 4 = 0'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.14
'(1) From the equation 2(6-x)=(x+2)(x-2) and x ≠ 2, we get x^2 + 2x - 16 = 0 and x ≠ 2.\n\nSolving this gives x = -1 ± √17\n\nSince this satisfies x ≠ 2, it is the desired solution.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.15
'Example: Solution to the equation z^n=α. Solve the equation z^3=4√3+4i. The solution can be in polar form.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.16
'(2) \ \\left|b_{n}-\\frac{1}{2}\\right|=\\frac{\\sqrt{5}}{2} \\quad \\cdots \\cdots \ (A) is proved by mathematical induction.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.17
'First, simplify the left side. Solve the equation 2 x-a=2 b-4 a+x by rearranging it.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.18
'(1) Let c be a real number. Determine the number of real solutions of the cubic equation x^{3}-3cx+1=0.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.19
'When 41m + n is even and 0, m + n is odd, \\\frac{2m}{m^{2}-n^{2}}\'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.20
'167 (2) y=\\frac{x^{3}}{2}+C x (where C is an arbitrary constant)'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.21
'(1) Let the volume of the cube after t seconds be V cm^3 and the surface area be S cm^2.\n\nAccording to the question, dV/dt=100\nAlso, the length of one side after t seconds is the cube root of V cm, so\nS=6(V^(1/3))^2=6 V^(2/3)\n\nTherefore, dS/dt=4 V^(-1/3) dV/dt=4/V^(1/3) * dV/dt\nSubstitute (1) and V^(1/3)=10 to get dS/dt=(4/10) * 100=40 cm^2/s\n()(a)Regarding the water volume after 5 seconds\n(pi/4)(h^2+h)=5 pi\nTherefore,\nh^2+h=20\nSince h>0, h=4 cm\n(b) Re-defining the height of the water surface after t seconds as h cm, then\n(pi/4)(h^2+h)=pi t\nSimplify to get h^2+h=4t\nDifferentiate both sides with respect to t to get (2h+1) dh/dt=4'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.22
'When a ≤ T, the equation x-1=\\sqrt{4 x^{2} - 4 x + a} has a real solution x=1. [Shibaura Institute of Technology]'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.23
'The equation of a certain line is y = 3x + 2. Find the coordinates of the point where this line intersects the line y = -2x + 5.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.24
'205 (1) y = 2x - 2, y = -6x + 22\n(2) y = 4, y = 9x - 14'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.25
'Please solve the quadratic equation x^2 + bx + c=0.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.26
'Find the range of values for the constant for which there exists a that satisfies the equation .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.27
'Find the condition for the equation to have solutions.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.29
'Determine the values of constants a and b such that the polynomial 2x^3 + ax^2 + bx - 3 is divisible by x - 3 and leaves a remainder of 5 when divided by 2x - 1.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.30
'Find the values of the constant for which all the solutions of the quadratic equation are integers, and determine all the integer solutions at that time.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.31
'When the cubic equation has three distinct real roots, find the range of the constant .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.32
"Eliminating y from (1) and (2), we get (α-β)/2 x=(α²-β²)/4, which simplifies to (α-β)/2 x=(α+β)(α-β)/4. Since α ≠ β, we have x=(α+β)/2. Substituting this into (1) gives y=α/2 * (α+β)/2-α²/4=αβ/4. Therefore, the coordinates of point P are (α+β)/2, αβ/4. From (3) we get αβ/4=-4/4=-1, hence y=-1. Conversely, when (4) holds, α, β are the two roots of the quadratic equation t²-2xt-4=0 with discriminant D', where D'/4=(-x)²-1*(-4)=x²+4. Thus, D' > 0, so there exist real numbers α, β (α ≠ β) for any x. Therefore, the trajectory of point P is the line y=-1."
A. ...
Ask AI tutor for answer!
Join our DiscordQ.34
'Given a+b+c=2, ab+bc+ca=3, abc=2, find the values of a^2+b^2+c^2 and a^5+b^5+c^5.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.35
'For real numbers a, b, and c that are not equal to 0 and satisfy a+b+c≠0 and abc≠0, if the equation 1/a + 1/b + 1/c = 1/(a+b+c) holds, then it is required to prove that for any odd number n, the equation 1/a^n + 1/b^n + 1/c^n = 1/(a+b+c)^n also holds.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.38
'Determine the types of solutions for the following quadratic equations. Here, k is a constant.\n(1) 3 x^{2}-5 x+3=0\n(2) 2 x^{2}-(k+2) x+k-1=0\n(3) x^{2}+2(k-1) x-k^{2}+4 k-3=0'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.39
'Find the coordinates of the intersection points between y=1 and y=\\frac{4}{3}x-\\frac{5}{3}.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.40
'Let be a quadratic polynomial. The polynomial does not divide , but is divisible by . Therefore, prove that the quadratic equation has a repeated root.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.41
'When x > 1, solving the mathematics II (2) problem: given the equation 4x^2 + 1/[(x+1)(x-1)] = 4(x^2-1) + 1/(x^2-1) + 4, under the conditions 4(x^2-1) > 0 and 1/(x^2-1) > 0, applying the relationship between arithmetic mean and geometric mean, we have the inequality 4(x^2-1) + 1/(x^2-1) + 4 ≥ 8. Therefore, it follows that the inequality 4x^2 + 1/[(x+1)(x-1)] ≥ 8 holds. The equality holds when 4(x^2-1) = 1/(x^2-1). Based on the equation (x^2-1)^2 = 1/4, and satisfying x > 1, we get x^2-1 = 1/2, hence x^2 = 3/2, which leads to x = sqrt(3/2) = sqrt(6)/2. Consequently, the minimum value of 4x^2 + 1/[(x+1)(x-1)] is 8, at x = sqrt(6)/2.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.42
'Find the range of values for the constant a, when at least one of the quadratic equations has no real roots.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.43
'Let a and b be real numbers. For the quadratic function f(x)=x^{2}+a x+b, answer the following questions.\n(1) When real numbers α, β satisfy f(α)=β, f(β)=α, α ≠ β, express α+β and αβ in terms of a, b.\n(2) Find the conditions on a, b for the existence of real numbers α, β satisfying f(α)=β, f(β)=α, α ≠ β.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.44
'Let a be a real number, and let the quadratic equation x^2+ax+(a-1)^2=0 have two distinct real roots. When the difference between the two roots is an integer, find the value of a.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.45
'If the three solutions of the cubic equation are denoted as , find the values of , , and .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.46
'When one of the solutions of the cubic equation is , find the values of the real constants and , and find the other solution.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.47
'Solutions and number of solutions of an nth degree equation'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.49
'Find the range of values for when the cubic equation has two different positive real solutions.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.50
'Let the three solutions of the cubic equation x^{3}-3 x^{2}-5=0 be α, β, γ. Find the following three numbers as solutions to a cubic equation: (1) α-1, β-1, γ-1 (2) β+γ/α, γ+α/β, α+β/γ'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.51
'Find all the values of when one of the two solutions of the quadratic equation is three times the other solution. The constraint is .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.54
'Determine the types of solutions for the following quadratic equations. Here, k is a constant.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.55
'Calculate the value of symmetric expressions of 2 solutions of a quadratic equation'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.56
'Solve the following equation for 0 ≤ θ < 2π. Also, find its general solution.\n(5) cos θ = 0'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.57
'For positive constants a, b, let f(x) = a*x^{2} - b. Prove that (1) f(f(x)) - x is divisible by f(x) - x. (2) Find the conditions for a, b so that the equation f(f(x)) - x = 0 has 4 distinct real solutions.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.58
'Please explain how to find the solutions of the quadratic equation ax^2+bx+c=0 and the discriminant D=b^2-4ac, and then discuss the difference in their results.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.59
'Prove that for real numbers x, y, z greater than or equal to 0, if x + y^2 = y + z^2 = z + x^2, then x = y = z.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.60
'Find the values of the constants a and b for which the function f(x)=x^{3}-a x^{2}+b has a maximum value of 5 and a minimum value of 1.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.61
'Find a complex number z such that squaring it results in i. z=x+yi (x, y are real numbers)'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.63
"Determine the range of values for the constant 'a' such that the quadratic equation has the following types of solutions:\n(1) Two distinct positive solutions\n(2) Solutions with opposite signs"
A. ...
Ask AI tutor for answer!
Join our DiscordQ.64
'The distance between the point (1,1) and the line ax-2y-1=0 is\n\n\\[ \\frac{|a \\cdot 1-2 \\cdot 1-1|}{\\sqrt{a^{2}+(-2)^{2}}} = \\frac{|a-3|}{\\sqrt{a^{2}+4}} \\]\n\nFrom the condition, \ \\frac{|a-3|}{\\sqrt{a^{2}+4}}=\\sqrt{2} \\nSquaring both sides, we get \\( \\frac{(a-3)^{2}}{a^{2}+4}=2 \\) therefore \\( (a-3)^{2}=2\\left(a^{2}+4\\right) \\)\nSolving, we have \ a^{2}+6 a-1=0 \ which gives \ a=-3 \\pm \\sqrt{10} \\nBoth sides are non-negative, so squaring is equivalent.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.65
'Practice\nDetermine the values of the constants a, b, c so that the equation \\( \\frac{1}{(x+1)(x+2)(x+3)}=\\frac{a}{x+1}+\\frac{b}{x+2}+\\frac{c}{x+3} \\) becomes an identity in x. [Similar to Shizuoka Institute of Science and Technology]\nBy multiplying both sides by \\( (x+1)(x+2)(x+3) \\), we obtain the equation\n1=a(x+2)(x+3)+b(x+1)(x+3)+c(x+1)(x+2)\nwhich is also an identity in x.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.66
'Let a>0, a≠1, b>0. Find all the points (a, b) on the coordinate plane for which the quadratic equation 4x^2+4x log_a b+1=0 has exactly one solution in the range 0<x<1/2.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.67
'Let ω be one of the solutions of the equation x² + x + 1 = 0. Express the value of f(ω) in terms of ω as a linear expression. Given that f(x) = x^80 - 3x^40 + 7.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.69
'Find the value of the real number k when the equation (i+1)x^2 + (k+i)x + ki + 1 = 0 has real roots, where i^2 = -1.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.70
'Let the three solutions of the cubic equation be . Then, the cubic equation with as solutions is: $x^{3}-5 x^{2}+8 x-3=0'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.71
'For a real constant p, when the quadratic equation x^{2}+px+p^{2}+p-1=0 has two distinct real solutions α and β, find the range of possible values for t=(α+1)(β+1).'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.72
'Determine the relationship and coefficients of the solution to 452\nFor the quadratic equation , find the value of the constant that satisfies the following conditions:\n(1) One root is twice the other root\n(2) One root is the square of the other root\nRefer to p. 78 Basic Concepts 1'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.73
'Solve the following equations, inequalities: [(1) Kyoto Sangyo University, (2) Jichi Medical University, (3) Seinan Gakuin University] (1) 8^x − 3 * 4^x − 3 * 2^{x+1} + 8 = 0 (2) 2(3^x + 3^-x) - 5(9^x + 9^-x) + 6 = 0 (3) 2^{x-4} < 8^{1-2x} < 4^{x+1}'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.74
'When the difference between the maximum and minimum values of the function f(x) = x^3 - 6x^2 + 3ax - 4 is 4, find the value of the constant a.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.75
'(1) Let the two solutions of the quadratic equation be denoted as . Find a quadratic equation whose solutions are .\n(2) Let the two different real number solutions of the quadratic equation be . It is known that the numbers are solutions to the quadratic equation . Find the values of the real constants and .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.76
'Let k be a constant. The line (k+3) x-(2 k-1) y-8 k-3=0 passes through a fixed point A independent of the value of k. Find the coordinates of that fixed point A.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.77
'Determine the range of the constant a, so that the equation x^{2}+ax+a=0 satisfies the following conditions:\n(1) Both solutions are less than or equal to 2.\n(2) One solution is greater than a, and the other solution is less than a.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.79
'Find the equations of the following lines: (1) passing through the point (-2,4) with slope -3 (2) passing through the point (5,6) parallel to the y-axis (3) passing through the point (8,-7) perpendicular to the y-axis (4) passing through the points (3,-5),(-7,2) (5) passing through the points (2,3),(-1,3) (6) passing through the points (-2,0),(0,3/4)'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.80
'Practice\n\n\nPractice (3) 159\nWhen \ 0 \\leqq \\theta \\leqq \\frac{\\pi}{2} \, solve the following equation.\n\\n\\cos \\theta+\\sqrt{3} \\cos 4 \\theta+\\cos 7 \\theta=0\n\\n\\[\egin{aligned}\n\\cos \\theta+\\sqrt{3} \\cos 4 \\theta+\\cos 7 \\theta & =\\\cos 7 \\theta+\\cos \\theta\+\\sqrt{3} \\cos 4 \\theta \\\\\n& =2 \\cos 4 \\theta \\cos 3 \\theta+\\sqrt{3} \\cos 4 \\theta \\\\\n& =\\cos 4 \\theta(2 \\cos 3 \\theta+\\sqrt{3})\n\\end{aligned}\\]\n\nTherefore, the equation is \\( \\quad \\cos 4 \\theta(2 \\cos 3 \\theta+\\sqrt{3})=0 \\)\nHence \ \\quad \\cos 4 \\theta=0 \ or \ \\cos 3 \\theta=-\\frac{\\sqrt{3}}{2} \\nSince \ 0 \\leqq \\theta \\leqq \\frac{\\pi}{2} \, we have \ 0 \\leqq 4 \\theta \\leqq 2 \\pi \, so within this range, \ \\cos 4 \\theta=0 \\nSolving this gives \ \\quad 4 \\theta=\\frac{\\pi}{2}, \\frac{3}{2} \\pi \, which means \ \\theta=\\frac{\\pi}{8}, \\frac{3}{8} \\pi \\nAlso, since \ 0 \\leqq 3 \\theta \\leqq \\frac{3}{2} \\pi \, within this range, solving \ \\cos 3 \\theta=-\\frac{\\sqrt{3}}{2} \ gives \ 3 \\theta=\\frac{5}{6} \\pi, \\frac{7}{6} \\pi \, i.e. \ \\theta=\\frac{5}{18} \\pi, \\frac{7}{18} \\pi \\n\n\ \\leftarrow Combine \\cos \\theta \ and \ \\cos 7 \\theta \.\n\ \\leftarrow \ Transform the left side from sum to product.\n\\( \\leftarrow Note the range of 4 \\theta.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.81
'Given that two of the solutions of the cubic equation x^3 + ax^2 - 21x + b = 0 are 1 and 3. Find the values of the constants a and b and the other solution.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.82
'Find the value of the constant when the ratio of the two solutions of the quadratic equation is .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.83
'Determine the types of solutions for the following 5 quadratic equations. Where k is a constant.\n(1) x^{2}-3 x+1=0\n(2) 4 x^{2}-12 x+9=0\n(3) -13 x^{2}+12 x-3=0\n(4) x^{2}-(k-3) x+k^{2}+4=0\n(5) x^{2}-(k-2) x+\\frac{k}{2}+5=0'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.84
"In the right division, if (remainder) = 0, then -8 + 4a = 0, -3a + b - 12 = 0, therefore a = 2, b = 18. At this point, the quotient is x + 6, so the equation is (x-1)(x-3)(x+6) = 0. Therefore, the other solution is x = -6 another solution 2. [(*) up to the same] Let's call the other solution c. Since f(x) also has a factor of x - c, the following identity holds for the next x. x³ + ax² - 21x + b = (x - 1)(x - 3)(x - c) Expanding and simplifying the right side yields x³ + ax² - 21x + b = x³ - (c + 4)x² + (4c + 3)x - 3c. Comparing the coefficients on both sides, we get a = -c - 4, -21 = 4c + 3, b = -3c. Solving this we get a = 2, b = 18; the other solution x = c = -6"
A. ...
Ask AI tutor for answer!
Join our DiscordQ.85
'Find the values of the constants for which the polynomial is divisible by . Next, factorize using the values of found within the real number range.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.86
'`a` is a real number, therefore `a=-3` which satisfies `a<0`. Hence, `a=3, b=5` or `a=-3, b=1`'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.87
'Let k be a real constant. Find the number of distinct real roots of the equation 2x^3 - 12x^2 + 18x + k = 0.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.88
'Let x>0, y>0, z>0. When 1/x + 2/y + 3/z = 1/4, find the minimum value of x + 2y + 3z.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.89
'When the equation x⁴ + ax² + b = 0 has 2 - i as a solution, find the values of the real constants a, b, and the other solution.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.90
'Solve the following equation and inequality when \ 0 \\leqq \\theta<2 \\pi \.อ161 (1) \ \\sin \\theta+\\sqrt{3} \\cos \\theta=\\sqrt{3} \ (2) \ \\cos 2 \\theta-\\sqrt{3} \\sin 2 \\theta-1>0 \'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.91
'For equations (1) and (2), find the range of values for the constant a that satisfy the following conditions:'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.92
'Let i be the imaginary unit. Let a and b be real numbers, and α be a non-real complex number. Given that α is a root of the equation x^2 - 2ax + b + 1 = 0, and α+1 is a root of the equation x^2 - bx + 5a + 3 = 0. Find the values of a, b, and α.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.93
'Let the two solutions of the quadratic equation be and . Then, , and .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.94
'Solve the following 2nd-degree equations:\n(1) \ 3 x^{2}+5 x-2=0 \\n(2) \ 2 x^{2}+5 x+4=0 \\n(3) \ \\frac{1}{10} x^{2}-\\frac{1}{5} x+\\frac{1}{2}=0 \\n(4) \\( (\\sqrt{3}-1) x^{2}+2 x+(\\sqrt{3}+1)=0 \\)'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.95
'Solve the quadratic equation to find the two roots as . Find the value of the following expressions.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.96
'Prove that for real numbers x, y, z greater than or equal to 0, if x+y^2=y+z^2=z+x^2, then x=y=z.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.97
'Prove the following for the three quadratic equations , , .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.98
'Create a quadratic equation with the following two solutions:'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.99
'In a factory, two types of products A and B are produced by two craftsmen, M and W. For product A, assembly work requires 6 hours per unit, and adjustment work requires 2 hours. Additionally, for product B, assembly work requires 3 hours, and adjustment work requires 5 hours. Either task can continue across days. Craftsman M is responsible only for assembly work, and craftsman W is responsible only for adjustment work. The time each craftsman spends on these tasks is limited to 18 hours per week for craftsman M and 10 hours per week for craftsman W. The goal is to maximize the total production of products A and B over 4 weeks. Find the total production quantity.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.01
'Solve the following equations, inequalities (0 ≤ θ < 2π):'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.02
'What is the method for determining the coefficients of a higher-degree equation?'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.03
'111 (1) (\\sqrt{3}-1)x+(\\sqrt{3}-1)y+\\sqrt{3}+1=0, (\\sqrt{3}+1)x-(\\sqrt{3}+1)y-\\sqrt{3}+1=0'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.04
'Find the range of values for the constant a when at least one of the two quadratic equations x^2 + 2ax + 4a - 3 = 0, 5x^2 - 4ax + a = 0 does not have real roots.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.05
'For the equations and , find the range of values for the constant that satisfy the following conditions (3). [Kurume University]'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.06
'Considering P = 2x^2 - (y - 5)x - (3y^2 + 5y - k) as a quadratic equation in x, from the quadratic formula we have x = (y - 5 ± √((y - 5)^2 + 8(3y^2 + 5y - k)))/4, in order for this to be expressed as a linear equation in y, the expression inside the square root must be a perfect square. Hence, the discriminant D of the condition 25y^2 + 30y + 25 - 8k = 0 should be 0, therefore find the value of k.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.07
'If this equation is an identity, then substituting x=2,1,-1 also holds true. Substituting these values, we get 3=3a, 4=-2b, 30=6c\nTherefore, a=1, b=-2, c=5\nIn this case, (right hand side) = x^{2}-1-2(x+1)(x-2)+5(x-2)(x-1)\n=x^{2}-1-2(x^{2}-x-2)+5(x^{2}-3x+2)\n=4x^{2}-13x+13\nand the given equation is an identity.\nTherefore, a=1, b=-2, c=5'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.09
'Find the discriminant of the quadratic equation ax^2 + bx + c = 0, and describe the nature of the solutions based on the value of the discriminant.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.10
'Find the value of α, which has two different roots α and α² for the quadratic equation x²-2kx+k=0 (where k is a constant).'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.11
'Proof: If a real coefficient n-th degree equation has an imaginary solution p+qi, then its conjugate complex number p-qi is also a solution.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.12
'Practice solving the following 2 quadratic equations.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.13
'Find the range of constant b for which the quadratic equation has real solutions regardless of the value of the constant a.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.14
'Let (a) be a constant. Find the range of values for (a) such that the equation has two distinct positive solutions.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.15
'Let \\\alpha, \eta, \\gamma\ be the three roots of the equation \x^3-2x^2-4=0\. Find the values of the following expressions.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.16
'Explain how to use the factor theorem to find one solution of a high-degree equation.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.17
'Example 161 (1)\n\\(\\sqrt {3} \\sin \\theta+\\cos \\theta+1=0 \n \\rightarrow 2 \\sin \\left(\\theta+\\frac{\\pi}{6}\\right)+1=0\\) is the normal equation'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.18
'Solve the following quadratic equations. (1) 4x^2-8x+3=0 (2) 3x^2-5x+4=0 (3) 2x(3-x)=2x+3 (4) 1/2x^2+1/4x-1/3=0 (5) (2+√3)x^2+2(√3+1)x+2=0'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.19
'Determine the value of the constant a so that 2x^{3} + 3ax^{2} - a^{2} + 6 is divisible by x + 1.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.20
'(3) Taking the common logarithm of both sides gives\\[ \\]\\nTherefore \\nFrom (1) we get \\nHence \\nTherefore \\nHere, \\nFind the coordinates included in the region represented by this system of inequalities.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.21
'Find the range of values for the constant b such that the equation x^2 - (8-a)x + 12-ab = 0 has real solutions regardless of the value of the real constant a.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.22
'For the two solutions α, β of the equation x^{2}-3 x+7=0, find the values of α^{2}+β^{2} and α^{4}+β^{4}. Also, find the value of (α^{2}+3 α+7)(β^{2}-β+7).'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.23
'If the two solutions of the quadratic equation 2x^2 - 4x + 1 = 0 are α and β, then α - 1/α, β - 1/β are the solutions of a new quadratic equation, create this new quadratic equation.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.24
'Prove the properties of a, b, and c satisfying the following conditions:'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.25
'Let a, b, p, q be real numbers. For the three quadratic equations x^2 + ax + b = 0 (1), x^2 + px + q = 0 (2), 2x^2 + (a+p)x + b+q = 0 (3), prove the following.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.27
'Given that two of the solutions of the cubic equation x^{3}+x^{2}+ax+b=0 are -1 and -3. Find the values of constants a and b, and determine the other solution.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.28
'Let the two solutions of the quadratic equation \ x^{2}-m x+p=0 \ be \ \\alpha, \eta \, and the two solutions of the quadratic equation \ x^{2}-m x+q=0 \ be \ \\gamma, \\delta \ (pronounced as Delta).\n(1) Express \\( (\\gamma-\\alpha)(\\gamma-\eta) \\) in terms of \ p, q \.\n(2) If \ p, q \ are the solutions of the equation \\( x^{2}-(2 n+1) x+n^{2}+n-1=0 \\), find the value of \\( (\\gamma-\\alpha)(\\gamma-\eta)(\\delta-\\alpha)(\\delta-\eta) \\).\n\ \\angle \ basic 39.46.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.29
'Determine the value of a such that the equation has a double root.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.30
'Translate the given text into multiple languages.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.32
'Practice solving the following equations and inequalities.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.33
'Practice (377 \ k \ is a constant. Determine the number of real solutions of the equation \ \\frac{2 x+9}{x+2}=-\\frac{x}{5}+k \.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.34
'Translate the given text into multiple languages.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.35
'What type of triangle is triangle ABC when three different points A(α), B(β), and C(γ) satisfy the following conditions?'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.36
'Practice. Find a complex number z such that its absolute value is 1 and z³-z is a real number.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.37
'Find the limit of the sequence determined by the following conditions. (2) 95\n(1) a1=2, an+1=3an+2\n(2) a1=1,2an+1=6-an'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.38
'The condition for one solution to be t<-1 and the other solution to be within the range -1<t is'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.39
'Find a fifth degree equation f(x) that satisfies conditions (A) and (B) simultaneously.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.40
'Find the function f(x) when the polynomial f(x) satisfies x f’’(x)+(1-x) f’(x)+3 f(x)=0, f(0)=1.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.41
'This is a math problem number \ \\mathbb{I} \\ -39 \. Find the range of existence for \ z \ that satisfies the following conditions.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.42
'Let {a_n} be a sequence with the sum from the first term to the nth term as S_n.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.44
'What is the minimum value of the exponent that satisfies the power equation?'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.45
'Let 3k be a real constant where k ≠ 0. The sequence {an} satisfying a1=1, k an + (2-k) an-1=1 (n=2,3,4,……) converges when the range of k is [], and lim{n→∞}an=1. (1) Find the general term an. (2) Find lim{n→∞}S(n)/an.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.47
'Let p and q be real numbers, with p^{2}-4q>0. Consider the equations x^{2}+px+q=0 and x^{2}-6x+13=0, where the real solutions of the first equation are α and β (α<β). Let the two complex solutions of the second equation be γ and δ (with γ having a positive imaginary part and δ having a negative imaginary part), and represent the points corresponding to α, β, γ, δ on the complex plane as A, B, C, D respectively. (1) Find the values of p and q when the quadrilateral AD【BC is a square. (2) Express q in terms of p when the four points A, B, C, D are on a single circumference.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.48
'Find all the solutions of the equation z³ + 3i z² - 3z - 28i = 0.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.49
'Investigate the number of real solutions of the equation 2√(x-1) = 1/2x + k based on the value of the constant k.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.50
'Find a complex number z with absolute value 1 such that z^3 - z is a real number.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.51
'Practice: What kind of shape does the set of points z satisfying the following equations form?'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.52
'Point P starts from vertex A of the square ABCD, and each time a die is rolled, it moves from one vertex to another according to the following rules.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.53
"Since x^{2}+x+1=\\left(x+\\frac{1}{2}\\right)^{2}+\\frac{3}{4}>0, inequality (1)'s each side multiplied by x^{2}+x+1 yields -\\left(x^{2}+x+1\\right)<1<x^{2}+x+1 -\\left(x^{2}+x+1\\right)<1 always holds. From 1<x^{2}+x+1 we get x^{2}+x>0 hence x(x+1)>0 thus x<-1,0<x Inequality (1) solution is x<-1,0<x Therefore, sum is x=-1, 0 f(x)=0 when x<-1, 0<x f(x)=\\frac{x^{2}+x}{1-\\frac{1}{x^{2}+x+1}}=x^{2}+x+1 therefore, the graph is as shown on the right. Therefore, continuous at x<-1,0<x, discontinuous at x=-1,0"
A. ...
Ask AI tutor for answer!
Join our DiscordQ.54
'Translate the given question to multiple languages.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.55
'Please plot the following shapes on the complex plane.④22\n(1) The locus of points \ z \ where \\\frac{1}{z+i}+\\frac{1}{z-i}\ is a real number, denoted as \ P \\n(2) When \ z \ moves along the locus \ P \ determined in (1), the locus of point \ w=\\frac{z+i}{z-i} \\n〔Hokkaido University〕 (1) \ z+i \\neq 0 \ and \ z-i \\neq 0 \ implies \\\quad z \\neq \\pm i \ and \\(\\quad \\frac{1}{z+i}+\\frac{1}{z-i}=\\frac{(z-i)+(z+i)}{(z+i)(z-i)}=\\frac{2z}{z^{2}+1}\\)\nWhen this is real\n\\[ \\frac{2z}{z^{2}+1}=\\overline{\\left(\\frac{2z}{z^{2}+1}\\right)} \\]\nFind the \ z \ that satisfies this condition.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.56
'In relation to the second-order curve F(x, y)=0 (A) and the line ax+by+c=0 (B), the coordinates of their common points are given by the real solutions of the simultaneous equations (A), (B). When the equation obtained by eliminating one variable from (A), B is a second-order equation, let its discriminant be D, then (1) D>0 (with different two sets of real solutions) ⇔ intersect at two points (2) D=0 (having one set of real solution [repeated]) ⇔ touching at one point (3) D<0 (having no real solution) ⇔ no common point. When the equation obtained by eliminating one variable from (A), B is a first-order equation (4) (having one set of real solution [non-repeated]) ⇔ intersect at one point. Based on these conditions, determine the situation of the common points between the second-order curve and the line.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.57
'Solve the following equations and inequalities. (1) \ \\sqrt{x+3}=|2 x| \ (2) \ \\sqrt{7 x-3} \\leqq \\sqrt{-x^{2}+5 x} \'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.58
'For α = 1, any number except -1; for α = -1, any number except 1; for α ≠ ±1, a number that satisfies |z| = 1 (where z ≠ -1/α)'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.60
'EXERCISES on Applications to Equations and Inequalities'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.61
'Solve the following equation and inequality: (1) \ \\sqrt{5-x}=x+1 \ (2) \ \\sqrt{5-x}<x+1 \'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.62
'By eliminating y from the equations of the straight line y=x+k and C2,'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.64
'From (1) x+y=1, we have y=-x+1\nSubstitute this into 2 x^{2}+y^{2}=1\nWe get 2 x^{2}+(-x+1)^{2}=1\nTherefore, 3 x^{2}-2 x=0\nThus x(3 x-2)=0\nTherefore, x=0 or \\frac{2}{3}\nHence, there are 2 intersection points\nFrom (2) 2 x-2 y+1=0, we have 2 x=2 y-1\nSubstitute this into y^{2}=2 x\nWe get y^{2}=2 y-1\nTherefore, y^{2}-2 y+1=0\nThus (y-1)^{2}=0\nTherefore, y=1\nHence, there is 1 intersection point'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.65
'Let k be a real constant. Find the number of solutions to the equation 4cos²x+3sinx-kcosx-3=0 for -π≤x≤π.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.66
'Solving the system of equations: -1=x-2, 2=y+3, -3=-z-4, we get x=1, y=-1, z=-1.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.67
'In the complex plane, let complex numbers represent the vertices of a triangle O, A, B as 0, α, β respectively.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.69
'When a complex number z = x + yi (where x, y are real numbers, and i is the imaginary unit) satisfies the following conditions, find the equation satisfied by x, y. Also, plot the general shape of the figure represented by the equation on the xy plane. (1) |z+3|+|z-3|=12'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.70
'Prove that the equation has at least one real solution.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.71
'The transformation from the expression to w using a first-order fraction conversion is called z.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.72
'What is the set of all points satisfying the following equations and inequalities:\n(1) \n(2) \n(3) \n(4) '
A. ...
Ask AI tutor for answer!
Join our DiscordQ.74
'Let n be a natural number greater than or equal to 2, consider the equation (1-x)e^{nx}-1=0. Where e is the base of natural logarithm.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.75
'Consider the sequence {an}. The sequence is defined by the following recurrence relation. a1 = 2, an+1 = pan + 2 (n = 1, 2, 3, ......). Find the general term of the sequence {an}. Furthermore, determine the range of values for p such that the sequence converges.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.76
'Find the number of real solutions of the equation 2 \\sqrt{x-1}=\\frac{1}{2} x+k. Here, k is a constant.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.78
'Determine the range of real numbers for which the sequence {((x^2+2x-5)/(x^2-x+2))^n} converges. Also, find the limit value at that point.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.79
'Find the number of distinct real solutions of the cubic equation ( is a constant).'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.80
'For points A(-3,0,4), B(x, y, z), C(5,-1,2), find the values of x, y, z that satisfy the following conditions: (1) The coordinates of the point that divides segment AB in the ratio 1:2 are (-1,1,3) (2) The coordinates of the point that divides segment AB in the ratio 3:4 are (-3,-6,4) (3) The coordinates of the centroid of triangle ABC are (1,1,3)'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.81
'Prove that when the fifth-degree equation ax^5 + bx^2 + c = 0 has a complex solution \ \\alpha \, its conjugate complex number \ \\overline{\\alpha} \ is also a solution.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.83
'Given two non-parallel vectors \ \\vec{a}, \\vec{b} \ (where, \ \\vec{a} \\neq \\overrightarrow{0}, \\vec{b} \\neq \\overrightarrow{0} \), when they satisfy the equation \\( 35 s(\\vec{a}+3 \\vec{b})+t(-2 \\vec{a}+\\vec{b})=-5 \\vec{a}-\\vec{b} \\), find the values of real numbers \ s \ and \ t \.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.84
'\\( y=-x,(-1,1) \\);\\[ y=-9 x+8,\\left(\\frac{1}{3}, 5\\right)\\]'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.85
'Consider the following equation:\n1. Equation: 3x^2 - 7y^2 - 6x + 24 = 0'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.86
'Translate the given text into multiple languages.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.87
'Find the coordinates (x, y, z) that satisfy the equation 2x + y - 3z - 4 = 0.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.88
'Prove that the inequality \\\left|\\frac{\\alpha-\eta}{1-\\overline{\\alpha} \eta}\\right|<1\ holds for complex numbers \\\alpha, \eta\ with absolute values less than 1. Here, \\\overline{\\alpha}\ denotes the complex conjugate of \\\alpha\.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.89
'A throws a dice in the first round, B throws in the second round, and then A and B take turns throwing the dice. The one who first throws a 1 or 2 wins. (1) Find the probability of A winning on the third round. (2) Let the probability of A winning by the th round be , find .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.90
'Prove the following theorem: Four points A(z_1), B(z_2), C(z_3), D(z_4) are concyclic if and only if the expression (z_2 - z_3)/(z_1 - z_3) divided by (z_2 - z_4)/(z_1 - z_4) is real.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.91
'Calculate the n-th power of a complex number (2) Complex number z satisfies z + 1/z = √2. (1) Express z in polar form. (2) Find the value of z^20 + 1/z^20. [Chubu University]'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.92
'(2) \ y = 4 x + \\sqrt{3} - \\frac{4}{3} \\pi \ English translation'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.93
'(1) When a = 2, d = -10, f = 0, the equation is 2x^2 + by^2 + cx - 10y = 0'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.94
'Let a, b, c be real numbers. Prove that when the 5th degree equation ax^5 + bx^2 + c = 0 has a complex solution α, its conjugate complex number α̅ is also a solution.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.95
'Let real numbers a, b be such that the cubic equation x^{3}+a x^{2}+b x+1=0 has an imaginary solution α. Prove that the conjugate complex number of α, ᾱ, is also a solution to this equation. Furthermore, express coefficients a, b, and the third solution β in terms of α and ᾱ.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.96
'Solution (1) Let \ \\sqrt{5-x}=x+1 \ be (*) Squaring both sides, we get \\[ 5-x=(x+1)^{2} \\] Simplifying, we get \ \\quad x^{2}+3 x-4=0 \ \\[ (x-1)(x+4)=0 \\] Solving this equation, we find \ \\quad x=1,-4 \ \ x=1 \ satisfies (*), but \ x=-4 \ does not. Thus, the solution is \ \\quad x=1 \ Additionally, since \ 5-x \\geqq 0 \ and \ x+1 \\geqq 0 \, we have \ -1 \\leqq x \\leqq 5 \ The solution to (**) is \ \\quad x=1 \'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.97
'Example 41 | Geometric Representation of Equations (1)What kind of shape does the set of all points z satisfying the following equations form? (1) |z+2i|=|z-3| (2) |z+1-3i|=2 (3) 4(z-1+i)(z̄-1-i)=1 (4) z+z̄=3'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.98
'Find the values of a, b, and c for Exercise 10 in Chapter 1 Functions.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.00
'Let a be a constant such that a > 1. Consider the function f(x) = \\frac{ax}{1 + ax}. (1) Prove that if a real number t satisfies f(f(t)) = f(t), then f(t) = t. (2) Solve the inequality f(f(x)) \\geqq f(x) for x. [Doshisha University] Example 11'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.01
'The quadratic equation for x with real numbers a, b has an imaginary solution z.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.05
'Solve the following equations and inequalities. (1) \ \\sqrt{2 x-1}=1-x \ (2) \ |x-3|=\\sqrt{5 x+9} \ (3) \ \\sqrt{3-x}>x-1 \ (4) \ x+2 \\leqq \\sqrt{4 x+9} \ (5) \ \\sqrt{2 x^{2}+x-6}<x+2 \'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.06
'The parabola , when translated parallel to the -axis by and parallel to the -axis by , touches both the lines and . Find the values of and .'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.07
'Please solve the conditions on the real number h: (1) x+y=h and (2) x/(y+4)=k, find the maximum value for each of them.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.08
'(1) Find the range of values for the constant so that the two quadratic equations have real roots simultaneously.\n(2) Find the value of the constant and the corresponding real root of the equation having exactly one real root.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.09
'From |3 x+2|=5, we get 3 x+2=±5, from 3 x+2=5 to 3 x+2=-5'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.10
'Let m be a real number constant. When the simultaneous linear equations in x and y are 2x+y-2=0 and mx-y-3m+1=0, and have a solution for x>0 and y>0, find the range of values for m.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.12
'Exercise 17 II (→ Textbook p.84) Investigate the relationship between condition P and conditions A ∼ F. ⟦ P → A ⟧ is true. (Counterexample: X = {n | n is a natural number}) ⟦ A → P ⟧ is clearly true. ⟦ P → B ⟧ is false. (Counterexample: X = {1}) ⟦ B → P ⟧ is false. (Counterexample: X = {n | n is an integer}) For ⟦ P → C ⟧, if the smallest number of elements in X exists, let it be x, then x-1 is an integer and x-1 is smaller than any element in X. Therefore, ⟦ P → C ⟧ is true.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.13
'Solve the equation: (3)(x+3)|x-4|+2(x+3)=0, which leads to (x+3)(|x-4|+2)=0. [1] When x >= 4, (1) becomes (x+3)(x-4+2)=0, so (x+3)(x-2)=0. Solving this gives x=-3, 2, but neither satisfies x >= 4. [2] When x < 4, (1) becomes (x+3){-(x-4)+2}=0, so (x+3)(-x+6)=0. Solving this gives x=-3, 6, and the only solution that satisfies x < 4 is x=-3. Combining [1] and [2], the solution is x=-3.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.14
'Solve the equation: Let x+1=t, then the equation becomes 6t^{2}+5t-14=0. Hence, (t+2)(6t-7)=0. Therefore, t=-2, 7/6. Substituting these into x=t-1 gives x=-3, 1/6.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.15
'Let a, b be different constants, and let the equations x^2+ax+ab^2=0 and x^2+bx+a^2b=0 have a common solution. When one of (1) and (2) has a double root, find the common solution. When neither (1) nor (2) has a double root, prove that at least one of the solutions is negative in the common set.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.16
'There are A ways to arrange 4 white balls, 3 black balls, and 1 red ball in a circle. Furthermore, there are B ways to create a ring by passing a string through these balls.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.17
'Solve the following equations or system of equations.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.18
'Practice 18\nBooklet p 53\n1. When x ≥ 4, the equation is x-4 = 3x\nSolving this, x=-2 which does not satisfy x ≥ 4.\n2. When x < 4, the equation is -(x-4) = 3x\nSolving this, x=1 which satisfies x < 4.\nFrom 1 and 2, the desired solution is x=1'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.19
'There are many cards of two types, white and black. When you have k cards in hand, consider the following operation (A): Select one card from the k cards in hand with equal probability of 1/k, and replace it with a card of a different color.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.20
'When α=-√2, from (1) we have 3-√2 a=0, therefore a=3/√2. In this case, the remaining solutions for (1) and (2) are 1/α,-1/α, which are -1/√2, 1/√2 respectively. Based on this, the number of desired intersection points is 2 when |a|<2, 3 when |a|=2, 4 when 2<|a|<3/√2, 3 when |a|=3/√2, and 4 when 3/√2<|a|. From alternative solution (*), we have a x=-x^2-1, a x=-3 x^2+3, therefore the number of desired intersection points is the same as the number of intersection points between the line y=a x and the curves y=-x^2-1 (1), y=-3x^2+3 (2). The discriminants of the quadratic equations a x=-x^2-1 and a x=-3 x^2+3 are denoted as D1 and D2 respectively, where D1=a^2-4=(a+2)(a-2), D2=a^2+36>0. The number of intersection points between the line y=a x and the curve (1) depends on D1: D1>0 gives 2 intersections when |a|>2, D1=0 gives 1 intersection when |a|=2, and D1<0 gives 0 intersections when |a|<2. Moreover, the line y=a x and the curve (2) always intersect at different two points. Furthermore, the x-coordinates of the intersection points of the curves (1) and (2) are obtained from -x^2-1=-3 x^2+3, resulting in x=±√2, thus the coordinates of the intersection points of the curves (1) and (2) are (±√2,-3). When the line y=a x passes through the point (±√2,-3), solving -3=±√2 a gives a=∓ 3/√2 (above, the signs are in the same order). From the above deductions, it is found that the results match those obtained from the diagram. It is known that 3/√2>3/1.5=2. When (1) has no real number solutions, there are 2 points. When (1) has repeated solutions, or when (1) and (2) have common solutions, there are 3 points. Otherwise, there are 4 points.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.22
'What steps should be taken to solve the equation 3x - 2 = 10?'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.23
'(2) \\\left\\{\egin{\overlineray}{l}x y+x=3 \\\\ 3 x y+y=8\\end{\overlineray}\\right.\'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.26
'Find the solution to the following system of equations.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.27
"\nExercise 33 III \ \\Rightarrow \ Book \ p.164 \\n(1) \\( f(x)=(x-1)^{2}+1, g(x)=-\\left(x-\\frac{a}{2}\\right)^{2}+\\frac{a^{2}}{4}+a \\)\n\nThe required condition is that \\( [f(x) \\) minimum value \\( ] \\geqq [g(x) \\) maximum value \ ] \\nTherefore \ 1 \\geqq \\frac{a^{2}}{4}+a \\nHence \ a^{2}+4 a-4 \\leqq 0 \\n\nSolving this gives \ -2-2 \\sqrt{2} \\leqq ~ a \\leqq-2+2 \\sqrt{2} \\n(2) \\( f(x)-g(x)=h(x) \\) and let\n\\[\n\egin{aligned}\nh(x) & =2 x^{2}-(a+2) x+2-a \\\n& =2\\left(x-\\frac{a+2}{4}\\right)^{2}-\\frac{1}{8}(a+2)^{2}+2-a \\\n& =2\\left(x-\\frac{a+2}{4}\\right)^{2}-\\frac{1}{8} a^{2}-\\frac{3}{2} a+\\frac{3}{2}\n\\end{aligned}\n\\]\nThe graph of \\( y=h(x) \\) is a concave parabola, axis is the line \ x=\\frac{a+2}{4} \\nThe required condition is that \\( h(x) \\geqq 0 \\) for \ 0 \\leqq x \\leqq 1 \\nTherefore, if the minimum value of \\( h(x) \\) in the range \ 0 \\leqq x \\leqq 1 \ is positive or zero, that's good.\n[1] \ \\frac{a+2}{4}<0 \ i.e., when \ a<-2 \, \\( h(x) \\) is minimum at \ x=0 \, and the minimum value is \\( h(0)=2-a \\)\nTherefore \ 2-a \\geqq 0 \\nSo \ a \\leqq 2 \\nThe common range with \ a<-2 \ is\n\ a<-2 \\n[2] \ 0 \\leqq \\frac{a+2}{4} \\leqq 1 \ i.e., when \ -2 \\leqq a \\leqq 2 \\n\\( h(x) \\) is minimum at \ x=\\frac{a+2}{4} \, and the minimum value is\n\\[h\\left(\\frac{a+2}{4}\\right)=-\\frac{1}{8} a^{2}-\\frac{3}{2} a+\\frac{3}{2} \\]"
A. ...
Ask AI tutor for answer!
Join our DiscordQ.28
'(2) \\( y=\\left|\\frac{1}{3}\\left(x^{2}+6 x-27\\right)\\right|=\\frac{1}{3}\\left|x^{2}+6 x-27\\right|=\\frac{1}{3}|(x+9)(x-3)| \\) \ x^{2}+6 x-27 \\geqq 0 \ The solution is, \\( (x+9)(x-3) \\geqq 0 \\) which gives\n\\nx \\leqq-9,3 \\leqq x\n\\n\ x^{2}+6 x-27<0 \ The solution is, \\( (x+9)(x-3)<0 \\) which gives\n\\n-9<x<3\n\'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.29
'(3) By the theorem of squares, x(x+5)=6² thus x²+5x-36=0, namely (x-4)(x+9)=0 since x>0, thus x=4'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.30
'(3) \ \\left\\{\egin{\overlineray}{l}x^{2}-y^{2}+x+y=0 \\\\ x^{2}-3 x+2 y^{2}+3 y=9\\end{\overlineray}\\right. \'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.32
'For the following two quadratic equations, find the range of values for the constant a that satisfy the conditions.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.33
'Practice moving a certain parabola symmetrically about the x-axis, then translating -1 in the x-axis direction and 2 in the y-axis direction, followed by symmetrical movement about the y-axis, resulting in the parabola y=-x^{2}-x-2. Find the equation of the original parabola.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.34
'Let a be a real number. For the quadratic equation in x, x^2 + (a+1)x + a^2 - 1 = 0, answer the following questions:'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.35
"The negation of the statement 'at least one does not have a real number solution' is 'both have real number solutions', therefore, the complement of range (1) is the range determined by (2). However, since \a \\neq 0\, the complement of \0 < a \\leqq \\frac{3}{2}\ is \a < 0, \\frac{3}{2} < a\."
A. ...
Ask AI tutor for answer!
Join our DiscordQ.36
'Determine the range of the constant a such that the quadratic equation ax^2-(a+1)x-2=0 has exactly one real root in the range -1<x<0 and 2<x<3.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.37
'Important Example 77 | Solutions of a Quadratic Equation and Comparing Coefficients'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.39
'Find the discriminant of the following quadratic equation and determine the number of roots based on the value of k.\n(1) x^{2}+2(k-4)x+k^{2}=0'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.40
"A few 150 yen snacks were purchased and put into a 200 yen box, with a total cost of 2000 yen. At this time, how many snacks were purchased? In such problems, unknown numbers are represented by letters. Let's denote the number of snacks as x, then the quantity relationship of the problem is represented by the following equation. 150x+200=2000"
A. ...
Ask AI tutor for answer!
Join our DiscordQ.41
'Solve the inequality: (1) Subtract 4 from each side and find the solution.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.43
'Find the intersection points of the following system of equations. \\\left\\{\egin{\overlineray}{l}y=x^{2}-4x \\\\ y=2x-9\\end{\overlineray}\\right.\'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.44
'Comprehensive exercise: Solve the simultaneous equations from (A) 4+2b+c=1, 25+5b+c=4 to find the values of b and c. Find the expression of f(x) and the vertex of the parabola at this time.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.45
'Solve the following problem: Let D be the discriminant of the quadratic equation (1), then D = (m - 1)^2 - 4n. The quadratic equation (1) has repeated roots when D = 0. Find the roots of equation (1). Also, find the solutions in the range 0 < x < 1.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.46
'(3) Let be the discriminant of the equation . Then , so there are no intersection points with the -axis.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.47
'(1) 35x + 91y + 65z = 3. By transforming (1), we get 7(5x + 13y) + 65z = 3. Let 5x + 13y = n, then 7n + 65z = 3. When n = 19 and z = -2, it is one of the integer solutions to (2). In this case, 5x + 13y = 19, x = -4, y = 3 is one of the integer solutions to this equation. Therefore, (x, y, z) = (-4, 3, -2) is one of the integer tuples that satisfies (1).'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.48
'Let a+b+c=8 (1) and a^{2}+b^{2}+c^{2}=32 (2). Substituting (1) into a=8-b-c, we get (8-b-c)^2+b^2+c^2=32. Simplifying, we have b^2+(c-8)b+(c-4)^2=0 (3) as the condition for real numbers b, c to satisfy. Therefore, the condition for the quadratic equation (3) to have real solutions is D≥0, where D is the discriminant of the quadratic equation (3). Simplifying D≥0 leads to -c(3c-16)≥0, which implies c(3c-16)≤0. Solving this inequality gives 0 ≥ c ≤ \\frac{16}{3}. Therefore, the maximum value of real number c is \\frac{16}{3}.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.49
'(1) Find the number of real solutions of the quadratic equation . Here, is a constant.\n(2) Determine the value of the constant such that the quadratic equation has repeated roots, and find those repeated roots.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.52
'Exercise 51 || We proceed from the textbook (p.244) (1). Since the average score of X is equal to the average score of Y, we have 3a + 3b + 4c = 5a + 5b, therefore c = (a + b) / 2. Please solve.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.54
'Find the common solution to the following equation.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.56
'There are many cards of two colors, black and white. When you have k cards in hand, consider the following operation (A).'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.57
'Common solution when k=0 is x=0; common solution when k=\x0crac{5}{22} is x=-\x0crac{1}{2}'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.58
'Let k be a constant. Find the number of distinct real solutions of the equation |x^2 - x - 2| - x + k = 0.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.59
'Find the solutions of the following equations:\n(1)\n\\n\egin{\overlineray}{l}\n\\frac{3x+2y}{4} = 1 \\\\\n\\frac{2x-y}{5} = 1\n\\end{\overlineray}\n\'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.60
'Find the number of real solutions for the following quadratic equations.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.61
'By utilizing the quadratic formula, x(x+9)=4⋅(4+5) leads to x²+9x-36=0, which simplifies to (x-3)(x+12)=0. Since x>0, x=3'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.63
'Practice finding a single common real solution for the following two quadratic equations by determining the value of the constant term as 58k for each:'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.64
'Since ax^{2}-3x+a=0 is a quadratic equation, a must not equal to 0. Find the discriminants D_{1} and D_{2} in each case, and determine the range of a.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.65
'Example 58 | Common Solution of Quadratic Equations (2)'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.66
'Given a line segment of length 1, draw a line segment with positive solutions to solve the following quadratic equations (1) x^2 + 4x - 1 = 0 (2) x^2 - 2x - 4 = 0'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.67
'Range of existence of solutions (relationship between solutions and the number k)\nLet\'s consider a quadratic function f(x)=a x^{2}+b x+c(a>0), where the quadratic equation f(x)=0 has two real solutions \\alpha, \eta(\\alpha \\leqq \eta), the relationship between the solutions \\alpha, \eta and the number k is as follows:\n(1) \\alpha>k, \eta>k\n(2) \\alpha<k, \eta<k\n(3) \\alpha<k<\eta\nWhen (1), (2) D=b^{2}-4 a c \\geqq 0, it indicates that there are intersection points with the x-axis. If f(k)>0, then the two points (\\alpha, 0), (\eta, 0) are both located to the right or left of the line x=k, but it is not yet determined which side is which, so the position of the axis is crucial. If the position of the axis is >k, then the two points (\\alpha, 0), (\eta, 0) are both to the right of the line x=k, i.e., \\alpha>k, \eta>k (see diagram (1)).\nIf the position of the axis is <k, then the two points (\\alpha, 0), (\eta, 0) are both to the left of the line x=k, i.e., \\alpha<k, \eta<k (see diagram (2)).\nIf f(k)<0, the graph will intersect with the x-axis in the ranges x<k and k<x.\nNote: The term "two real solutions" here does not just refer to two different real solutions, it also includes the case of double roots. However, the third case does not include the case of double roots.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.68
'What is the condition for the equation h(x)=mx to have three distinct real number solutions?'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.69
'When the proposition p ➡️ q is true, q is said to be a necessary condition for p, and is a sufficient condition for q. This relationship can be remembered in the following form.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.70
'For the quadratic equation , determine the range of the constant such that the equation has real solutions satisfying the following conditions:\n1. All solutions are greater than 1.\n2. One solution is greater than 1, and the other solution is less than 1.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.71
'Practice (2) Let a, b, c, d be positive numbers. For the quadratic equation x^2-(a+b)x+ab-cd=0, answer the following questions. [Shinshu University] (A) Prove that it has two distinct real solutions. (B) Prove that at least one of the two solutions must be positive. (C) Let the two solutions be α, β, and assume 0<α<β, investigate the relationship between a, a+b, α, β.'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.72
'Example 39 ⇒ Book p.364 (1) By the power rule, x⋅6=3⋅4 therefore x=2'
A. ...
Ask AI tutor for answer!
Join our DiscordQ.77
"Let's practice making 600g of 5% saltwater using 20 exercise books, page 55. First, establish equations for the amounts of saltwater and salt. Since there are conditions for x and y, use these conditions to derive an inequality involving only z. Take x grams and y grams from 3% and 4% saltwater solutions respectively, add z grams of 7% saltwater to create 600g of 5% saltwater. The total amount of saltwater is x+y+z=600. Next, for the amount of salt, set up the equation x × 3/100 + y × 4/100 + z × 7/100 = 600 × 5/100."
A. ...
Ask AI tutor for answer!
Join our DiscordQ.78
"A group of elementary school students decided to purchase books and donate them to a public facility, with the target amount set at a yen. It was decided that each person would contribute 60 yen, and if all members of the group, b people, contributed, it was expected to exceed the target amount. However, because 20% of the members did not contribute, the target amount was short by 980 yen. Consequently, it was decided that only those who contributed money would each contribute an additional 15 yen, but as 3 people did not contribute, the target amount was still not reached. Nevertheless, the shortfall of c yen was covered by the bookstore's service, enabling the planned donation of books. Find the values of a, b, and c."
A. ...
Ask AI tutor for answer!
Join our DiscordQ.79
A product sells 100 units per day when the unit price is 10 yen. For each 1 yen increase in the unit price, daily sales decrease by 5 units, and for each 1 yen decrease in the unit price, daily sales increase by 5 units. At what unit price will the daily sales amount be maximized? Find the maximum daily sales amount and the unit price at that time. Note that consumption tax is not considered.
A. ...
Ask AI tutor for answer!
Join our DiscordQ.80
Solve the linear equation using the properties of equality.
A. ...
Ask AI tutor for answer!
Join our DiscordQ.81
Solve the following quadratic equations.
(1)
(2)
(3) rac{1}{2} x^{2}+rac{2}{3} x-1=0
A. ...
Ask AI tutor for answer!
Join our DiscordQ.82
Solving a quadratic equation using the basic 87 formula
A. ...
Ask AI tutor for answer!
Join our DiscordQ.84
For the three quadratic functions
\((1), y=(x-2)^{2}\)\( (2), y=(x-2)^{2}+1 \)
, the correspondence table of and is as follows.
A. ...
Ask AI tutor for answer!
Join our DiscordQ.85
Example 37
Determine the number of apples based on the conditions of the price, quantity, and total cost of 1 unit of apples and oranges.
A. ...
Ask AI tutor for answer!
Join our DiscordQ.86
Given the function \( f(x)=x^{2}+a x+b \), express the minimum value over the interval in terms of and .
A. ...
Ask AI tutor for answer!
Join our DiscordQ.87
Choose the appropriate option for the following! from the options (1) to (3) below. Assume x is a real number.
(1) p: x^{2}-x=0 \quad q: x=1 then, p is the \square condition for q.
(2) For quadrilateral p: rhombus q: diagonals intersect perpendicularly then, p is the \square condition for q.
① Necessary and sufficient condition
② Necessary condition but not sufficient
③ Sufficient condition but not necessary
A. ...
Ask AI tutor for answer!
Join our DiscordQ.88
TRAINING 88
(1) Find the range of values for the constant when the quadratic equation has no real solutions.
(2) Find the value of the constant and the repeated root for the quadratic equation \( x^{2}-2 m x+2(m+4)=0 \) when it has a repeated root.
A. ...
Ask AI tutor for answer!
Join our DiscordQ.89
Condition for the quadratic equation to have 88 real roots (1)
A. ...
Ask AI tutor for answer!
Join our DiscordQ.90
Standard 74 | Maximum and Minimum of Quadratic Functions and Word Problems (1)
A. ...
Ask AI tutor for answer!
Join our DiscordQ.91
State the converse and the contrapositive of the following proposition and determine their truth values. Also, state the inverse of proposition . Assume are real numbers and is an integer.
(1) implies at least one of or is a negative number.\」
(2) P: \left\lceil n^{2}+1
ight. is even is odd
(3) P: \left\lceil 3 x+5>0 \Longrightarrow x^{2}-6 x-7=0
ight
floor
A. ...
Ask AI tutor for answer!
Join our DiscordQ.92
(1) Remove the absolute value signs of .
(2) Solve the equation .
A. ...
Ask AI tutor for answer!
Join our DiscordQ.93
Basic 63 | Function Expressions and Function Values
A. ...
Ask AI tutor for answer!
Join our DiscordQ.94
An equation of the form (quadratic expression of x) = 0 is called a quadratic equation. The values of x that satisfy this quadratic equation are called the solutions of the quadratic equation, and finding all solutions is called solving the quadratic equation. Here, let us examine the solutions of quadratic equations, including what was learned in middle school.
A. ...
Ask AI tutor for answer!
Join our DiscordQ.95
Solve the following quadratic equations.
(1)
(2)
(3)
(4)
(5)
(6)
A. ...
Ask AI tutor for answer!
Join our DiscordQ.96
Determine the values of constants and to satisfy the following condition.
(1) The graph of the linear function passes through the two points \( (-2,2) \) and \( (4,-1) \).
A. ...
Ask AI tutor for answer!
Join our DiscordQ.97
Solve the following quadratic equations.
(1)
(2)
(3)
(4)
(5)
A. ...
Ask AI tutor for answer!
Join our DiscordQ.00
TRAINING 56
Let x and y be real numbers. State the converse, contrapositive, and inverse of the following propositions, and determine their truth values.
(1) x^2 ≠ -x ⟹ x ≠ -1 (2) x+y is a rational number ⟹ x or y is a rational number
A. ...
Ask AI tutor for answer!
Join our DiscordQ.01
When the quadratic equation has a repeated root, find the value of the constant and its repeated root.
A. ...
Ask AI tutor for answer!
Join our DiscordQ.02
Determine the value of the constant such that the minimum value of the function \( f(x)=-x^{2}+4 x+c(-4 \leqq x \leqq 4) \) is -50.
A. ...
Ask AI tutor for answer!
Join our DiscordQ.05
Therefore, \overrightarrow{\mathrm{DH}}=rac{1}{30} k ec{b}+rac{1}{5} k ec{c}-rac{9}{10} k ec{d} , and since is on the plane , there exist real numbers such that \overrightarrow{\mathrm{AH}}=s ec{b}+t ec{c} . Hence, \overrightarrow{\mathrm{DH}}=\overrightarrow{\mathrm{AH}}-\overrightarrow{\mathrm{AD}}=s ec{b}+t ec{c}-ec{d} . From (1) and (2), we get rac{1}{30} k ec{b}+rac{1}{5} k ec{c}-rac{9}{10} k ec{d}=s ec{b}+t ec{c}-ec{d}. Since the 4 points are not on the same plane, rac{1}{30} k=s, \quad rac{1}{5} k=t, \quad-rac{9}{10} k=-1
A. ...
Ask AI tutor for answer!
Join our DiscordQ.06
The straight line y=2x+k intersects the ellipse x^{2}+4y^{2}=4 at two distinct points P and Q. (1) Find the range of the constant k. (2) When k varies within the range found in (1), find the locus of the midpoint M of the line segment PQ.
A. ...
Ask AI tutor for answer!
Join our DiscordQ.08
Find the equation of the circle with center at point and radius .
A. ...
Ask AI tutor for answer!
Join our DiscordQ.09
Math question: For , let . Determine the range of existence for point when the real numbers and satisfy the following conditions:
(1)
(2)
A. ...
Ask AI tutor for answer!
Join our DiscordQ.11
White Chart Method is a technique used in solving algebraic equations. Suppose you have the equation x^2 - 5x + 6 = 0.
A. ...
Ask AI tutor for answer!
Join our DiscordQ.12
51 (1) \( r \cos \left( heta-rac{\pi}{4}
ight)=\sqrt{2} \)
(2) \( r=2 \sqrt{2} \cos \left( heta-rac{\pi}{4}
ight) \)
A. ...
Ask AI tutor for answer!
Join our DiscordQ.14
Using the White Chart Method, solve the equation x^2 + x - 12 = 0.
A. ...
Ask AI tutor for answer!
Join our DiscordQ.16
What shape do all the points that satisfy the following equations form?
(1) \( |z|^{2}=2i(z-ar{z}) \)
(2)
A. ...
Ask AI tutor for answer!
Join our DiscordQ.17
White Chart Method is a technique used in solving algebraic equations. Suppose you have the equation x^2 - 5x + 6 = 0.
A. ...
Ask AI tutor for answer!
Join our DiscordQ.18
Using the White Chart Method, solve the equation x^2 + x - 12 = 0.