Relations between roots and coefficients - One Line Questions
1.
If $\alpha$ and $\beta$ are the roots of $x^2 + 2x + 3 = 0$, find the value of $\frac{1}{\alpha^2} + \frac{1}{\beta^2}$. —
-2/9
2.
If $\alpha$ and $\beta$ are the roots of $x^2 + 2x + 4 = 0$, find the value of $\alpha^2 \beta + \alpha \beta^2$. —
-4
3.
If $\alpha$ and $\beta$ are the roots of $x^2 + px + q = 0$, then the quadratic equation whose roots are $\alpha + 1$ and $\beta + 1$ is: —
$(x-1)^2 + p(x-1) + q = 0$
4.
If the roots of the quadratic equation $x^2 - px + q = 0$ are $\alpha$ and $\beta$, and the roots of $x^2 - rx + s = 0$ are $\alpha + \delta$ and $\beta + \delta$, then what is $r-p$? —
2$\delta$
5.
If $\alpha$ and $\beta$ are the roots of $ax^2 + bx + c = 0$, then the value of $\alpha^2 \beta + \alpha \beta^2$ is: —
$\frac{-bc}{a^2}$
6.
If $\alpha$ and $\beta$ are the roots of $ax^2 + bx + c = 0$, then the value of $(\alpha - \beta)^2$ is: —
$\frac{b^2 - 4ac}{a^2}$
7.
For a quadratic equation $ax^2 + bx + c = 0$, where $a \neq 0$, if $\alpha$ and $\beta$ are the roots, what is the sum of the roots? —
$\frac{-b}{a}$
8.
For a quadratic equation $ax^2 + bx + c = 0$, where $a \neq 0$, if $\alpha$ and $\beta$ are the roots, what is the product of the roots? —
$\frac{c}{a}$
9.
If $\alpha$ and $\beta$ are the roots of $ax^2 + bx + c = 0$, then the value of $(\alpha + 1)(\beta + 1)$ is: —
$\frac{c+b+a}{a}$
10.
If $\alpha$ and $\beta$ are the roots of $x^2 - px + q = 0$, then the value of $\frac{\alpha}{\beta} + \frac{\beta}{\alpha}$ is: —
$\frac{p^2 - 2q}{q}$
11.
If $\alpha$ and $\beta$ are the roots of $x^2 - 4x + 3 = 0$, find the equation whose roots are $\frac{1}{\alpha}$ and $\frac{1}{\beta}$. —
$3x^2 - 4x + 1 = 0$
12.
If $\alpha$ and $\beta$ are the roots of $ax^2 + bx + c = 0$, then the quadratic equation whose roots are $\frac{1}{\alpha}$ and $\frac{1}{\beta}$ is: —
$cx^2 + bx + a = 0$
13.
If the roots of $x^2 - px + q = 0$ are $\alpha$ and $\beta$, then the value of $\alpha^2 + \beta^2 + \alpha \beta$ is: —
$p^2 - 3q$
14.
If $\alpha$ and $\beta$ are the roots of $x^2 - px + q = 0$, then the value of $(\alpha + \beta)^2$ is: —
$p^2$
15.
If $\alpha$ and $\beta$ are the roots of $x^2 - px + q = 0$, then the value of $(\alpha \beta)^2$ is: —
$q^2$
16.
If the roots of $x^2 + px + q = 0$ are $\alpha$ and $\beta$, then the equation whose roots are $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$ is: —
$q x^2 + (p^2 - 2q)x + q = 0$
17.
If $\alpha$ and $\beta$ are the roots of $x^2 - 2x + 4 = 0$, find the equation whose roots are $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$. —
$x^2 - (-4)x + 1 = 0$
18.
If $\alpha$ and $\beta$ are the roots of $x^2 + px + q = 0$, then the equation whose roots are $\alpha^2$ and $\beta^2$ is: —
$x^2 - (p^2 - 2q)x + q^2 = 0$
19.
If $\alpha$ and $\beta$ are the roots of $x^2 - 5x + 6 = 0$, find the equation whose roots are $\alpha^2$ and $\beta^2$. —
$x^2 - 13x + 36 = 0$
20.
If the roots of $x^2 - 6x + 5 = 0$ are $\alpha$ and $\beta$, find the equation whose roots are $\alpha^2$ and $\beta^2$. —
$x^2 - 26x + 25 = 0$
21.
Let $\alpha$ and $\beta$ be the roots of the quadratic equation $ax^2 + bx + c = 0$. If $\alpha + \beta = 2$ and $\alpha \beta = -3$, find the quadratic equation. —
$x^2 - 2x - 3 = 0$
22.
If $\alpha$ and $\beta$ are the roots of $x^2 - 5x + 6 = 0$, find the equation whose roots are $\alpha-1$ and $\beta-1$. —
$x^2 - 3x + 2 = 0$
23.
If $\alpha$ and $\beta$ are the roots of $x^2 - 3x + 2 = 0$, find the equation whose roots are $\alpha+2$ and $\beta+2$. —
$x^2 - 7x + 10 = 0$
24.
If the roots of $x^2 + ax + b = 0$ are $\alpha$ and $\beta$, then the equation whose roots are $\alpha^2$ and $\beta^2$ is: —
$x^2 - (a^2 - 2b)x + b^2 = 0$
25.
If $\alpha$ and $\beta$ are the roots of $x^2 + px + q = 0$, then the equation whose roots are $\alpha^3$ and $\beta^3$ is: —
$x^2 - (p^3 - 3pq)x + q^3 = 0$
26.
If $\alpha$ and $\beta$ are the roots of $x^2 - 2x + 3 = 0$, find the equation whose roots are $\alpha^3$ and $\beta^3$. —
$x^2 + 6x + 27 = 0$
27.
If $\alpha$ and $\beta$ are the roots of $x^2 + bx + c = 0$, then the equation whose roots are $\alpha - k$ and $\beta - k$ is: —
$x^2 + (b-2k)x + (c-bk+k^2) = 0$
28.
If $\alpha$ and $\beta$ are the roots of $x^2 + x + 1 = 0$, find the equation whose roots are $\alpha^2$ and $\beta^2$. —
$x^2 + x + 1 = 0$
29.
If the sum of the roots of the quadratic equation $kx^2 - 6x - 2 = 0$ is 3, find the value of $k$. —
2
30.
If $\alpha$ and $\beta$ are the roots of $x^2 - 5x + 6 = 0$, find the value of $(\alpha - \beta)^2$. —
1
31.
If $\alpha$ and $\beta$ are the roots of $x^2+x+1=0$, then the value of $\alpha^3+\beta^3$ is: —
-1
32.
If $\alpha$ and $\beta$ are the roots of $x^2+x+1=0$, then the value of $\alpha^4+\beta^4$ is: —
-1
33.
If $\alpha$ and $\beta$ are the roots of $x^2 + x + 1 = 0$, find the value of $\alpha^2 + \beta^2$. —
-1
34.
If the roots of $x^2 - 5x + 6 = 0$ are $\alpha$ and $\beta$, then the value of $\alpha^2 + \beta^2 + \alpha \beta$ is: —
7
35.
If $\alpha$ and $\beta$ are the roots of $x^2 - 6x + 8 = 0$, then the value of $\alpha^3 + \beta^3$ is: —
144
36.
If $\alpha$ and $\beta$ are the roots of $x^2 - 7x + 12 = 0$, then the value of $\alpha^3 + \beta^3$ is: —
189
37.
Consider the quadratic equation $2x^2 + 4x + 3 = 0$. If its roots are $\alpha$ and $\beta$, what is $\alpha + \beta$? —
-2
38.
Consider the quadratic equation $2x^2 + 4x + 3 = 0$. If its roots are $\alpha$ and $\beta$, what is $\alpha \beta$? —
3/2
39.
If $\alpha$ and $\beta$ are the roots of $x^2 - 5x + 6 = 0$, find the value of $\frac{\alpha}{\beta} + \frac{\beta}{\alpha}$. —
13/6
40.
If the product of the roots of the quadratic equation $3x^2 + kx - 5 = 0$ is -5/3, find the value of $k$. —
5
41.
If one root of the quadratic equation $x^2 - kx + 8 = 0$ is 2, find the value of $k$. —
5
42.
If $\alpha$ and $\beta$ are the roots of $x^2 - 3x + 2 = 0$, find the value of $\frac{1}{\alpha} + \frac{1}{\beta}$. —
3/2
43.
If $\alpha$ and $\beta$ are the roots of $x^2 + 2x + 3 = 0$, find the value of $(\alpha - \beta)^2$. —
-4
44.
If one root of the quadratic equation $2x^2 + kx - 3 = 0$ is -3, find the value of $k$. —
5
45.
If the roots of the quadratic equation $x^2 - 5x + 6 = 0$ are $\alpha$ and $\beta$, find $\alpha + \beta$. —
5
46.
If the roots of the quadratic equation $x^2 - 5x + 6 = 0$ are $\alpha$ and $\beta$, find $\alpha \beta$. —
6
47.
If $\alpha$ and $\beta$ are the roots of $x^2 - 3x + 2 = 0$, find the value of $\alpha^2 + \beta^2$. —
7
48.
If $\alpha$ and $\beta$ are the roots of $x^2 - 3x + 2 = 0$, find the value of $\frac{1}{\alpha^2} + \frac{1}{\beta^2}$. —
5/4
49.
If $\alpha$ and $\beta$ are the roots of $x^2 - 2x + 3 = 0$, find the value of $\alpha^3 + \beta^3$. —
-6
50.
If $\alpha$ and $\beta$ are the roots of $x^2 - 3x + 5 = 0$, find the value of $(\alpha - 1)(\beta - 1)$. —
7