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