# 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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.09

'For the equation $x^{2}-2(k-3) x+4 k=0$, determine the range of constant $k$ so that the equation has the following roots:'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.10

'Solve the following system of simultaneous equations.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.12

'Determine the types of solutions for the following quadratic equations.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.13

'Practice solving the following equations and inequalities.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.14

'Find the values of the constant $m$ for which the quadratic equation $x^2 - mx + 3m = 0$ has only integer solutions, and determine all such integer solutions.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.15

'Find the sum and product of the two solutions of the following quadratic equations.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.16

'Solve the following practice problem: Find the solutions to the quadratic equation.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.17

'Important Example 23 | Solutions of Quadratic Equations and the Value of Equations Let the two solutions of the quadratic equation $x^{2}+n x+p=0$ be $a, b$, and let the two solutions of $x^{2}+n x+q=0$ be $c, d$. Here, $p, q$ are integers, and $n$ is a real number. (1) Express $(c-a)(c-b)$ in terms of $p, q$. (2) Prove that $(a-c)(b-d)(a-d)(b-c)$ is a perfect square (can be expressed as the square of an integer).'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.22

'Solve trigonometric equations using sum and product formulas.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.25

'Based on the following conditions, solve the problem.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.28

'(1) Let $\\alpha$ and $\eta$ be the two solutions of the quadratic equation $x^{2}+3x-6=0$. Determine a new quadratic equation with solutions $2\\alpha+\eta$ and $\\alpha+2\eta$. (2) If $\\alpha$ and $\eta$ are the solutions of the quadratic equation $x^{2}+px+q=0$, and one of the quadratic equations with solutions $\\alpha^{2}$ and $\eta^{2}$ is $x^{2}-4x+36=0$, find the values of the real constants $p$ and $q$.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.30

'Find all integer values of $m$ for which the quadratic equation $x^{2} + (2m + 5)x + m + 3 = 0$ has integer solutions.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.32

'Translate the given text into multiple languages.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.33

'(1) (6, 4) (2) In order (4x + 3y -17 = 0, 3x - 4y + 6 = 0)'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.40

'Find the general term of the sequence {an} defined by the following conditions.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.51

'For the quadratic equation $a x^{2}+b x+c=0$ with two solutions $\\alpha, \eta$, we have $\\alpha+\eta=-\\frac{b}{a}$ and $\\alpha \eta=\\frac{c}{a}$.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.52

'Let $D_{1}$, $D_{2}$, and $D_{3}$ 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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.62

'Solve the following 4th degree equation: x^{4}=4'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.65

'Let $Q(x)$ be a quadratic polynomial. The polynomial $P(x)$ cannot divide $Q(x)$, but {$P(x)$}^2 can be divided by $Q(x)$. Prove that the quadratic equation $Q(x)=0$ has a repeated root.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.66

'Since the real number is positive, it follows that $x-2>0$ and $3-x>0$. Therefore, $2<x<3$. And since $2\\log_{4}(3-x)=\\log_{2^{2}}(3-x)^{2}=\\log_{2}(3-x)$.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.67

'Interpret this as solving for b in the 2nd degree equation'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.68

'Using the relationship between the roots and coefficients, find the following value.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.69

'Important Example 27 | Solutions of Two Equations'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.71

'For all natural numbers n, derive cn + 1 by utilizing an + bn + cn = 1.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.76

'a³ - a² - b = 0 or 9a + 27b - 1 = 0 where a ≠ 1/3'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.77

'The base is a positive number that is not equal to 1.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.78

'Find the condition for one of the equations to have complex roots.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.80

'Find the quadratic equation using the sum and product of two numbers.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.81

'Example 18 The value of the symmetric equation (2)\nFor the two roots $\\alpha, \eta$ of the 2nd degree equation $2 x^{2}+4 x+3=0$, find the values of the following expressions.\n(1) $\\alpha^{5}+\eta^{5}$\n(2) $(\\alpha-1)^{4}+(\eta-1)^{4}$'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.82

'Example 38 Recurrence Relation Among Adjacent 3 Elements (2)'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.90

'Let the discriminants of the equations (1) and (2) be denoted as D1 and D2.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.92

'Since the point (1,2) lies on the line (3), we have a+2b=1'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.95

'Show the relationship between the solutions of a cubic equation and the coefficients.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.00

'Find the value of k that satisfies the following conditions.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.03

'Given mathematical text converted to multiple languages.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.05

'When the equation $x^{2}+y^{2}-4 k x+(6 k-2) y+14 k^{2}-8 k+1=0$ represents a circle\n(1) Find the range of values for the constant $k$.\n(2) When $k$ varies within this range, find the trajectory of the center of the circle.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.09

'Find the number of distinct real solutions to the following cubic equations.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.10

'Confirm logarithmic equations and conditions of exponents'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.11

'When the cubic equation $x^{3}-(a+2) x+2(a-2)=0$ has a double root, find the value of the constant $a$.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.13

'Find the two numbers that have the sum and product as follows:'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.14

'Develop 52: Proof problem regarding the solutions of a quadratic equation'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.18

'When the quadratic equation $x^{2}+2(a+3) x-a+3=0$ has two different solutions both greater than 1, find the range of values for the constant $a$.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.19

'For the quadratic equation $ax^2 + bx + c = 0$ with two solutions $\\alpha, \eta$ and discriminant $D$:\n1. $\\alpha, \eta$ are two distinct positive solutions $\\Longleftrightarrow D > 0$ and $\\alpha + \eta > 0$ and $\\alpha \eta > 0$\n2. $\\alpha, \eta$ are two distinct negative solutions $\\Longleftrightarrow D > 0$ and $\\alpha + \eta < 0$ and $\\alpha \eta > 0$\n3. $\\alpha, \eta$ are solutions with opposite signs $\\quad \\Longleftrightarrow \\alpha \eta < 0$'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.21

'Find the number of distinct real solutions for the following cubic equations.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.22

'Determine the range of values for the constant $m$ such that the quadratic equation $x^{2}+2mx+6-m=0$ has two distinct real roots both greater than 1.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.23

'Determine the range of values for the constant m so that the quadratic equation $x^2 + 2(m-1)x + 2m^2 - 5m - 3 = 0$ 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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.26

'Higher degree equation: Find the value of the constant $a$ and the other root of the equation $x^3-ax^2+(3a-1)x-24=0$, given that one of the roots is $x=2$.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.28

'For the quadratic equation $4 x^{2}+4(m+2) x+9 m=0$, answer the following questions.\n(1) Determine the range of values for the constant $m$ when the equation has two complex solutions.\n(2) Find the values of the constant $m$ and the repeated root when the equation has a repeated root.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.33

'If the three solutions of the cubic equation $x^{3}+x^{2}+x+3=0$ are denoted as $α, β, γ$, find the values of $α^{2}+β^{2}+γ^{2}$ and $α^{3}+β^{3}+γ^{3}$.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.34

'When the cubic equation $x^{3} + (a+1)x^{2} - a = 0$ has a repeated root, find the value of the constant $a$.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.35

'Find the sum and product of the two solutions to the following quadratic equations.\n(1) $x^{2}-4 x-3=0$\n(2)$2 x^{2}-3 x+6=0$\n(3) $3 x^{2}=5-4 x$'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.37

'Basic 62: Solving Higher Degree Equations (2) - Utilizing the Factor Theorem'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.44

'Find the general term of the sequence defined by the recurrence relation $a_{1}=1, a_{n+1}=2a_n+3^n$.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.48

'Find the range of values for the constant $a$ when the quadratic equation $x^{2}-a x+4=0$ has two distinct solutions, both of which are less than 3.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.49

'Extension 53: Integer solutions of quadratic equations (using the relationship between solutions and coefficients)'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.50

'Solve the high degree equation x^{3}-4 x^{2}+2 x+4=0.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.51

'Basic Example 62 Determine Coefficients of Degree 64 Polynomial (1) ... Conditions for Real Solutions 3rd degree equation $x^{3}+ax^{2}-17x+b=0$ has -1 and -3 as solutions. (1) Find the values of constants $a$ and $b$. (2) Find the other solutions to this equation.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.52

'When the quadratic equation $x^{2}-2 a x+3 a-2=0$ has two distinct positive solutions, find the range of values for the constant $a$.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.55

'Find the formula for the solutions of the quadratic equation ax^{2}+bx+c=0.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.56

'Find the value of the constant $m$ and the two solutions when the two solutions of the quadratic equation $3 x^{2}+6 x+m=0$ satisfy the following conditions:'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.58

'Consider the cubic equation $x^{3}+(a-5)x^{2}+(a+8)x-6a-4=0$.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.59

"Find information about 'high-degree equations' based on the following table."

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.62

'Find the value of the constant m and the two solutions of the quadratic equation $3x^{2}+6x+m=0$ 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 Discord#### Q.64

'Find the equation of a circle passing through three points.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.65

'Development 69: Solution of Higher Order Equations (3)'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.69

'Find the range of values for the constant $a$ when the equation $a\\left(x^{2}-x+1\\right)=1+2 x-2 x^{2}$ has real solutions.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.71

'The function $y=4^{x}+4^{-x}-2^{x+1}-2^{1-x}$ takes the minimum value $b$ at $x=a$. Find the value of $|a+b|$.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.74

'Investigate the conditions for a cubic equation to have repeated roots'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.76

'Basic 43: Value of two solutions of a symmetric equation'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.80

'Basic 42: Sum and product of the two solutions to a quadratic equation'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.86

'The condition for having only imaginary solutions is'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.92

'Development 54: Range of existence of solutions of a quadratic equation (2)'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.95

'Find the range of values for the constant $a$ when the cubic equation $x^{3}-3 a^{2} x+4 a=0$ has three distinct real roots.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.96

'Consider the quadratic equation $x^{2}+(m+1)x+m-1=0$.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.97

'Standard 65: Determining coefficients of higher order equations (2) - Conditions for imaginary solutions'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.98

'Find the value of the constant $a$ when the cubic equation $x^{3}-(a+2) x+2(a-2)=0$ has a double root.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.99

'Standard 49: Range of existence of solutions of a quadratic equation (1)'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.01

'Basic 41: Conditions for a quadratic equation to have complex roots, repeated roots'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.02

'Let the three solutions of the cubic equation $x^{3}-2 x+1=0$ be $\\alpha, \eta, \\gamma$. Find the value of the following expressions.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.07

'Standard 40: Discrimination of types of solutions of quadratic equations (2)'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.09

"Determine the range of the constant 'm' so that the quadratic equation $x^{2}+2(m-1)x+2m^{2}-5m-3=0$ 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 Discord#### Q.10

'Expansion 66: Relationship between the solutions of a cubic equation and its coefficients'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.14

'Find the sum and product of the two roots of the following quadratic equations.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.16

'Find two numbers whose sum is 2 and product is -4.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.17

'Find the range of values for the constant $a$ when the cubic equation $x^{3}-3a^{2}x+4a=0$ has three distinct real roots.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.18

'Developing 68: Conditions for a cubic equation to have three distinct real roots'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.27

'Find a fifth-degree polynomial f(x) that satisfies conditions (A) and (B) simultaneously.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.29

'Find the velocity, acceleration, position, and distance traveled (linear motion).'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.31

'Fundamentals 8: Algebraic solutions for irrational equations and inequalities'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.32

'For a point $(a, b)$ on the hyperbola $x^{2}-4 y^{2}=4$ with a tangent line having a slope $m$, answer the following questions. Assume $b \neq 0$.\n(1) Find the relationship between $a, b, m$.\n(2) Let the distance between a point on this hyperbola and the line $y=2x$ be denoted as $d$. Find the minimum value of $d$. Also, determine the coordinates of the point on the curve that provides the minimum value of $d$.[Kanagawa University]'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.33

'What geometric shape is formed by the set of points that satisfy the following equations?'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.45

'Solve the following quadratic equations:\n(1) $x^{2}-3 x+2=0$\n(2) $2 x^{2}-3 x-35=0$\n(3) $12 x^{2}+16 x-3=0$\n(4) $14 x^{2}-19 x-3=0$\n(5) $5 x^{2}-3=0$\n(6) $(2 x+1)^{2}-9=0$'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.49

'Find the range of values for the constant a that satisfy the given conditions for the two quadratic equations $x^{2}-x+a=0$ and $x^{2}+2ax-3a+4=0$.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.52

'Solve the following system of simultaneous equations.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.53

'How many ways are there to divide 12 different books as follows?'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.55

'If the two distinct real solutions of the quadratic equation $x^{2}-a^{2}x-4a+2=0$ are denoted as $\\alpha$ and $\eta$, and satisfy $1<\\alpha<2<\eta$, determine the range of values of the constant $a$.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.56

'When the equation $ax^{2}+bx+1=0$ has two solutions $-2,3$, find the values of constants $a, b$.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.58

'Determine the number of real solutions of the quadratic equation $2 x^{2}+3 x+k=0$.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.59

'Solve the following system of simultaneous equations.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.63

'What is the range of existence of the roots of a quadratic equation?'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.64

'Find the number of real solutions for the following quadratic equations.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.66

'Find the range of values for the constant $a$ such that the quadratic equation $x^{2}+a x-a^{2}+a-1=0$ has two distinct real solutions within the range $-3<x<3$.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.67

'Determine the range of values for the constant $a$ when the quadratic equation $x^{2}-2 a x+a+6=0$ satisfies the following conditions: (1) Has positive and negative roots. (2) Has two distinct negative roots.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.69

'Transform the given mathematical expression into a different form.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.72

'Please state the converse, contrapositive, and inverse of the proposition.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.73

'Determining coefficients from maximum and minimum values (3)'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.76

'What is the range of values for $k$ for which the quadratic equations $x^{2}+x+k=0$, $x^{2}+k x+1=0$ both have real number solutions?'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.78

'Use the quadratic formula to solve the following 2nd-degree equations.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.80

'Determine the truth values of the following propositions.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.82

'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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.85

'Solve the following system of simultaneous equations.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.89

'Solve the following system of simultaneous equations.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.91

'Basic Example 30 Number of Integer Solutions (Using Combinations with Repetition)'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.95

"Please provide 'Linear Diophantine Equations' and its page."

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.99

'Solve equations and inequalities involving absolute value: (1) $|x|=c$, (2) $|x|<c$, (3) $|x|>c$'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.02

'For the function $f(x)=x^{2}+2x-a^{2}+5$, find the range of values of $a$ that satisfy the following conditions:'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.04

'When the quadratic equation $x^2 - 2mx + 2(m + 4) = 0$ has equal roots, find the value of the constant $m$ and the equal roots at that time.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.11

'For the quadratic equation $x^{2}+mx+9=0$, find the range of values for the constant $m$ when it has the following:'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.14

'When the solutions of the quadratic equation $x^{2}+2 m x-m+2=0$ are as follows, find the range of the constant $m$. (1) Having two distinct real solutions. (2) Having real solutions. (3) Having no real solutions.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.19

'Determine the range of values for the constant $a$ so that the quadratic equation $x^{2}-(a-4)x+a-1=0$ 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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.22

'When the roots of the quadratic equation $x^{2}+m x+9=0$ have the following characteristics, find the range of values for the constant $m$.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.25

'Translate the given text into multiple languages.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.26

'Find the value of the constant $m$ and the repeated root of the quadratic equation $x^{2}-2mx+2(m+4)=0$ when it has repeated roots.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.27

'When the quadratic equation $x^{2}+5x+7-m=0$ has two distinct real roots, find the range of values for the constant $m$.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.30

'(2) The two inequalities $x^{2}+mx+m+2<0$ have solutions'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.33

'The classification of solutions of a quadratic equation can be determined by examining the sign of the discriminant $D = b^{2} - 4ac$.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.36

'Find the values of $a, b, c$ when the 3rd-degree function $f(x)=x^{3}+a x^{2}+b x+c$ satisfies $(x-2)f^{prime}(x)=3 f(x)$.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.38

'(1) Find the solutions of the equation 2x^2 - 2√6x + 3 = 0.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.41

'Find the value of a that satisfies the given equations.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.42

'Prove that the equation $ax+by+cz=0$ holds true when (2) $\\frac{x}{b-c}=\\frac{y}{c-a}=\\frac{z}{a-b}$.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.44

'The solutions to the cubic equation $x^{3}+a x^{2}-21 x+b=0$ are $1,3, c$. Find the values of constants $a$, $b, c$.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.45

'Find the value of a that satisfies the following equation. a=2'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.49

'When the equation $x^{2}+y^{2}-6 k x+(12 k-2) y+46 k^{2}-16 k+1=0$ represents a circle,'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.52

'Find all real numbers $a$ such that the cubic equation $x^{3}+(a+2)x^{2}-4a=0$ has exactly 2 real roots.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.54

'After reading the conversation between Hanako and Taro about the problem, answer the following question.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.56

'Transform the given text into multiple languages.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.62

'If the quadratic equation $x^{2}+4 a x+8 a^{2}-20 a+25=0$ has real roots, then $a=\\{ \\text{constant} \\}$ and the solutions are $x=\\{ \\text{constant} \\}$.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.63

'Create a quadratic equation with the sum as \ p \ and the product as \ q \.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.64

'Find the values of real numbers x, y that satisfy the following system of equations.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.66

'Create a quadratic equation with the roots α and β.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.68

'When the equation $x^{4}-x^{3}+a x^{2}+b x+6=0$ has roots at $x=-1,3$, find the values of constants $a, b$. Also, find the other roots in this case.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.69

'The sum and product of the two roots of the quadratic equation $x^{2}+a x+b=0$ are the roots of the quadratic equation $x^{2}+b x+a=0$. Determine the values of constants $a, b$.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.76

'Basic Column Question 18 Determining the Coefficients of an Identity'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.77

'Use recurrence relation to find the general term of the sequence.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.78

'Given the quadratic equation $3 x^{2}-2 x-4=0$ with two roots $\\alpha, \eta$, find the value of the following expressions.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.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 Discord#### Q.89

'Example 76: Equation of a line passing through a fixed point'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.91

'Determine the types of solutions for the following quadratic equations:'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.92

'Solve the system of equations x^2-3xy+y^2=19, x+y=2.'

#### A. ...

##### Ask AI tutor for answer!

Join our Discord#### Q.93

'When the equation $x^{2}+y^{2}+6 p x-2 p y+28 p+6=0$ represents a circle, find the range of values for the constant $$