Algebra and Quadratic Equations - Question Bank

1. If x^2 + 1/x^2 = 11, what is the value of x + 1/x (assuming x is positive)?
A) 3
B) sqrt(11)
C) sqrt(13)
D) 4
2. If x = 3, what is the value of x^3 - 3x^2 + 3x - 1?
A) 7
B) 8
C) 9
D) 10
3. What is the nature of the roots of the quadratic equation x^2 + x + 1 = 0?
A) Real and distinct
B) Real and equal
C) Complex conjugates
D) One real, one imaginary
4. If x^2 - 36 = 0, what are the values of x?
A) 6
B) -6
C) 6, -6
D) 36
5. Solve for x: 2x^2 + 5x - 3 = 0
A) x = 1/2, x = -3
B) x = -1/2, x = 3
C) x = 3/2, x = -1
D) x = -3/2, x = 1
6. If x^2 + 1/x^2 = 22, and x is positive, find the value of x + 1/x.
A) 4
B) 5
C) 6
D) 7
7. If x^2 - 8x + 16 = 0, what is the value of x?
A) 4
B) -4
C) 8
D) 16
8. If x^2 = 100, what is the value of x?
A) 10
B) -10
C) 10, -10
D) 20
9. What is the smallest integer value of x for which x^2 - 5x + 6 < 0?
A) 0
B) 1
C) 2
D) 3
10. If x + 1/x = 5, what is the value of x^3 + 1/x^3?
A) 110
B) 125
C) 130
D) 140
11. Find the roots of the quadratic equation x^2 - 7x + 10 = 0.
A) x = 2, x = 5
B) x = -2, x = -5
C) x = 2, x = -5
D) x = -2, x = 5
12. If x^3 = 64, what is the real value of x?
A) 4
B) 8
C) 16
D) 64
13. If x + y = 11 and x^2 + y^2 = 61, find the value of xy.
A) 20
B) 25
C) 30
D) 35
14. If x^2 = 49, what is the value of x?
A) 7
B) -7
C) 7, -7
D) 49
15. What is the sum of squares of the roots of x^2 - 5x + 6 = 0?
A) 13
B) 25
C) 36
D) 5
16. If x^2 - 10x + 25 = 0, what is the value of x?
A) 5
B) -5
C) 25
D) 10
17. Solve the equation: (x - 2)(x + 3) = 0
A) x = 2, x = -3
B) x = -2, x = 3
C) x = 2, x = 3
D) x = -2, x = -3
18. If x + 1/x = 4, what is the value of x^2 + 1/x^2?
A) 14
B) 16
C) 18
D) 20
19. Find the product of the roots of x^2 + 8x + 15 = 0.
A) 8
B) -8
C) 15
D) -15
20. If x^3 = 27, what is the value of x?
A) 3
B) 9
C) 81
D) 27
21. If x^2 + 2x + 1 = 0, what is the value of x?
A) 1
B) -1
C) 0
D) 2
22. What is the value of x^2 - y^2 if x + y = 10 and x - y = 4?
A) 14
B) 40
C) 60
D) 100
23. If x^2 + 1/x^2 = 47, find the value of x - 1/x (assuming x > 1).
A) 5
B) 6
C) 7
D) 8
24. What are the roots of the quadratic equation 3x^2 + 5x - 2 = 0?
A) x = 1/3, x = -2
B) x = -1/3, x = 2
C) x = 2/3, x = -1
D) x = -2/3, x = 1
25. If x - 1/x = 3, what is the value of x^3 - 1/x^3?
A) 18
B) 27
C) 36
D) 54
26. If x^2 - 4 = 0, what is the value of x^2?
A) 2
B) 4
C) 8
D) 16
27. Solve for x: x^2 - 6x + 9 = 0
A) x = 3
B) x = -3
C) x = 3, x = -3
D) x = 9
28. If x + 1/x = 2, what is the value of x^n + 1/x^n for any positive integer n?
A) 2
B) 0
C) 1
D) n
29. Find the sum of the roots of the equation x^2 - 11x + 30 = 0.
A) 11
B) -11
C) 30
D) -30
30. If x^3 - 1 = 0, what is the real value of x?
A) 1
B) -1
C) 0
D) 3
31. What is the value of x if x^2 = 16?
A) 4
B) -4
C) 4, -4
D) 8
32. If x^2 + y^2 = 25 and x + y = 7, find the value of xy.
A) 6
B) 12
C) 18
D) 24
33. If x^2 - 5x + 6 = 0, what is the value of x(x-5)?
A) -6
B) 6
C) 5
D) -5
34. Find the roots of the quadratic equation x^2 + 4x + 4 = 0.
A) x = 2
B) x = -2
C) x = 2, x = -2
D) x = 4
35. If x^2 + 1/x^2 = 18, and x is positive, find the value of x + 1/x.
A) 4
B) 5
C) 6
D) 7
36. If x = 5, what is the value of x^2 - 2x + 1?
A) 16
B) 20
C) 24
D) 36
37. What is the smallest root of the equation x^2 - 4x + 3 = 0?
A) 1
B) 3
C) -1
D) -3
38. If x^3 = 8, what is the real value of x?
A) 2
B) -2
C) 4
D) 1
39. Solve for x: 3x^2 - 10x + 7 = 0
A) x = 1, x = 7/3
B) x = -1, x = -7/3
C) x = 1, x = -7/3
D) x = -1, x = 7/3
40. If x + y = 7 and xy = 12, find the value of x^2 + y^2.
A) 25
B) 30
C) 35
D) 40
41. If x^2 - 9 = 0, what are the values of x?
A) 3
B) -3
C) 3, -3
D) 9
42. What is the discriminant of the quadratic equation ax^2 + bx + c = 0?
A) b^2 - 4ac
B) b^2 + 4ac
C) 4ac - b^2
D) sqrt(b^2 - 4ac)
43. If x = 2 is a root of the quadratic equation x^2 - kx + 8 = 0, find the value of k.
A) 4
B) 5
C) 6
D) 8
44. For the quadratic equation 2x^2 - 7x + 3 = 0, find the nature of the roots.
A) Real and distinct
B) Real and equal
C) Complex
D) One real, one complex
45. If x - 1/x = 5, then what is the value of x^2 + 1/x^2?
A) 23
B) 25
C) 27
D) 29
46. What is the product of the roots of the quadratic equation ax^2 + bx + c = 0?
A) -b/a
B) c/a
C) b/a
D) -c/a
47. If a quadratic equation has roots 3 and -5, what is the equation?
A) x^2 + 2x - 15 = 0
B) x^2 - 2x - 15 = 0
C) x^2 + 2x + 15 = 0
D) x^2 - 2x + 15 = 0
48. What is the sum of the roots of the quadratic equation ax^2 + bx + c = 0?
A) c/a
B) -b/a
C) b/a
D) -c/a
49. Solve the quadratic equation: x^2 - 5x + 6 = 0
A) x = 1, x = 6
B) x = 2, x = 3
C) x = -2, x = -3
D) x = -1, x = -6
50. If x + 1/x = 3, then what is the value of x^2 + 1/x^2?
A) 5
B) 7
C) 9
D) 11
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