Quadratic Equations - Question Bank

1. If the roots of the equation x^2 - 7x + k = 0 are 3 and 4, what is the value of k?
A) 12
B) -12
C) 7
D) -7
2. Which of the following quadratic equations has roots 1/2 and -1/3?
A) 6x^2 + x - 1 = 0
B) 6x^2 - x - 1 = 0
C) 6x^2 + 5x - 1 = 0
D) 6x^2 - 5x - 1 = 0
3. What is the discriminant of the quadratic equation 2x^2 + 5x - 3 = 0?
A) 25
B) 49
C) -25
D) -49
4. If one root of the equation x^2 - (a+1)x + 6 = 0 is 2, find the other root.
A) 3
B) 4
C) 5
D) 6
5. What is the value of k if the equation x^2 - 4x + k = 0 has roots that are equal?
A) 4
B) -4
C) 2
D) -2
6. If the roots of the equation x^2 - 5x + k = 0 are 2 and 3, what is the value of k?
A) 6
B) -6
C) 5
D) -5
7. The equation 3x^2 - 6x + 2 = 0 has roots α and β. What is the value of αβ?
A) 2/3
B) -2/3
C) 6/3
D) -6/3
8. The equation 3x^2 - 6x + 2 = 0 has roots α and β. What is the value of α + β?
A) 2
B) -2
C) 6
D) -6
9. If one root of the equation x^2 - 3x + k = 0 is 1, what is the value of k?
A) 2
B) -2
C) 3
D) -3
10. What is the value of k for which the roots of the equation x^2 - (k+2)x + 3k = 0 are equal?
A) 3
B) 6
C) 9
D) 12
11. If the sum of the roots of a quadratic equation is S and the product is P, the equation can be written as:
A) x^2 + Sx + P = 0
B) x^2 - Sx + P = 0
C) x^2 + Px + S = 0
D) x^2 - Px + S = 0
12. Consider the quadratic equation x^2 - 7x + 10 = 0. What is the difference between its roots?
A) 1
B) 2
C) 3
D) 5
13. If the roots of the equation x^2 + bx + c = 0 are equal, then:
A) b^2 = 4c
B) b^2 = -4c
C) c^2 = 4b
D) c^2 = -4b
14. The equation x + 1/x = 5 can be written as a quadratic equation. What is it?
A) x^2 - 5x + 1 = 0
B) x^2 + 5x + 1 = 0
C) x^2 - 5x - 1 = 0
D) x^2 + 5x - 1 = 0
15. If one root of the equation ax^2 + bx + c = 0 is the square of the other, what is the relation between a, b, and c?
A) b^3 = a(c - ab)
B) b^3 = c(a - bc)
C) b^3 = a^2c + ac^2
D) b^3 + ac + a^2c^2 = 0
16. What is the quadratic equation whose roots are reciprocals of the roots of x^2 - 5x + 6 = 0?
A) 6x^2 - 5x + 1 = 0
B) 6x^2 + 5x + 1 = 0
C) 5x^2 - 6x + 1 = 0
D) x^2 - 6x + 5 = 0
17. The product of the roots of the equation (x-2)(x+3) = 0 is:
A) 1
B) -1
C) 5
D) -6
18. The sum of the roots of the equation (x-2)(x+3) = 0 is:
A) 1
B) -1
C) 5
D) -5
19. If one root of the equation ax^2 + bx + c = 0 is 0, then which coefficient must be zero?
A) a
B) b
C) c
D) a and b
20. What is the value of k if the quadratic equation x^2 - kx + 4 = 0 has equal roots?
A) ±4
B) ±2
C) ±8
D) ±16
21. For the equation x^2 + 4 = 0, what are the roots?
A) 2 and -2
B) 2i and -2i
C) 2i and 2i
D) 4i and -4i
22. If the roots of x^2 - ax + b = 0 are m and n, then mn equals:
A) b
B) -b
C) a
D) -a
23. If the roots of x^2 - ax + b = 0 are m and n, then m+n equals:
A) b
B) -b
C) a
D) -a
24. The equation 3x^2 + 2x - 1 = 0 has roots:
A) 1/3 and -1
B) -1/3 and 1
C) 1/3 and 1
D) -1/3 and -1
25. Which method involves rewriting the quadratic expression as a perfect square?
A) Factoring
B) Quadratic formula
C) Completing the square
D) Graphical method
26. The product of the roots of the equation 2x^2 - 7x + 6 = 0 is:
A) 7/2
B) -7/2
C) 3
D) -3
27. The sum of the roots of the equation x^2 + 5x + 3 = 0 is:
A) 5
B) -5
C) 3
D) -3
28. If one root of the quadratic equation x^2 - (k+1)x + 6 = 0 is 2, find the other root.
A) 3
B) 4
C) 5
D) 6
29. What is the value of k for which the equation kx^2 - 5x + k = 0 has equal roots?
A) ±√(5/2)
B) ±5/2
C) ±2/5
D) ±√(2/5)
30. If the roots of ax^2 + bx + c = 0 are a and b, then which of the following is true?
A) a + b = -b/a and ab = c/a
B) a + b = b/a and ab = -c/a
C) a + b = -c/a and ab = b/a
D) a + b = c/a and ab = -b/a
31. What is the nature of the roots of the equation 3x^2 - 5x + 2 = 0?
A) Real and distinct
B) Real and equal
C) Complex conjugate
D) No real roots
32. What is the nature of the roots of the equation 4x^2 - 12x + 9 = 0?
A) Real and distinct
B) Real and equal
C) Complex conjugate
D) No real roots
33. What is the nature of the roots of the equation x^2 + x + 1 = 0?
A) Real and distinct
B) Real and equal
C) Complex conjugate
D) One real, one complex
34. If x = 2 is a root of 2x^2 - kx + 6 = 0, what is the value of k?
A) 7
B) -7
C) 8
D) -8
35. The equation (x+1)^2 = 9 is a quadratic equation. What are its roots?
A) 2 and -4
B) -2 and 4
C) 3 and -3
D) 1 and -1
36. Which quadratic equation has roots -1/2 and 1/3?
A) 6x^2 + x - 1 = 0
B) 6x^2 - x - 1 = 0
C) 6x^2 + 5x - 1 = 0
D) 6x^2 - 5x - 1 = 0
37. Which quadratic equation has roots 5 and -2?
A) x^2 - 3x - 10 = 0
B) x^2 + 3x - 10 = 0
C) x^2 - 3x + 10 = 0
D) x^2 + 3x + 10 = 0
38. If the roots of the quadratic equation x^2 - px + 12 = 0 are 3 and 4, what is the value of p?
A) 7
B) -7
C) 12
D) -12
39. If one root of the quadratic equation x^2 - 6x + k = 0 is 2, what is the value of k?
A) 8
B) -8
C) 6
D) -6
40. For the quadratic equation ax^2 + bx + c = 0, what is the product of the roots?
A) -b/a
B) b/a
C) c/a
D) -c/a
41. For the quadratic equation ax^2 + bx + c = 0, what is the sum of the roots?
A) c/a
B) -b/a
C) b/a
D) -c/a
42. What are the roots of the quadratic equation x^2 - 2x + 5 = 0?
A) 1 + 2i and 1 - 2i
B) 2 + i and 2 - i
C) 1 + i and 1 - i
D) 2 and -1
43. What are the roots of the quadratic equation x^2 + 4x + 4 = 0?
A) 2 and 2
B) -2 and -2
C) 4 and 1
D) -4 and -1
44. What are the roots of the quadratic equation x^2 - 5x + 6 = 0?
A) 2 and 3
B) 1 and 6
C) -2 and -3
D) 1 and 5
45. If the discriminant (D = b^2 - 4ac) of a quadratic equation is negative (D < 0), what type of roots does it have?
A) Two distinct real roots
B) Two equal real roots
C) No real roots (two complex conjugate roots)
D) One real root
46. If the discriminant (D = b^2 - 4ac) of a quadratic equation is zero (D = 0), what type of roots does it have?
A) Two distinct real roots
B) Two equal real roots
C) No real roots
D) One complex root
47. If the discriminant (D = b^2 - 4ac) of a quadratic equation is positive (D > 0), what type of roots does it have?
A) Two equal real roots
B) Two distinct real roots
C) No real roots (two complex roots)
D) One real root
48. What is the discriminant of a quadratic equation ax^2 + bx + c = 0?
A) b^2 - 4ac
B) b^2 + 4ac
C) -b + sqrt(b^2 - 4ac) / 2a
D) (-b +/- sqrt(b^2 - 4ac)) / 2a
49. Which of the following is a method to solve quadratic equations?
A) Graphical method
B) Completing the square
C) Factoring
D) All of the above
50. In a quadratic equation ax^2 + bx + c = 0, what condition must 'a' satisfy?
A) a = 0
B) a != 0
C) a > 0
D) a < 0
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