Quadratic Equations

1. What is the standard form of a quadratic equation?
  a) ax + b = 0
  b) ax² + bx + c = 0
  c) ax² = c
  d) ax³ + bx² + c = 0
  Answer: b
2. In a quadratic equation, what must be true about the coefficient of
  x2x^2
  (a)?
  a) a = 0
  b) a < 0
  c) a ≠ 0
  d) a > 0
  Answer: c
3. If the roots of the quadratic equation are real and equal, what is the discriminant?
  a) Greater than 0
  b) Less than 0
  c) Equal to 0
  d) None of the above
  Answer: c
4. Which of the following equations is quadratic?
  a) x + 2 = 5
  b)
  x2+3x−4=0x^2 + 3x – 4 = 0
  c)
  3x3+2x+1=03x^3 + 2x + 1 = 0
  d)
  2x−4=02x – 4 = 0
  Answer: b
5. The roots of the quadratic equation
  x2−5x+6=0x^2 – 5x + 6 = 0
  are:
  a) 2 and 3
  b) -2 and 3
  c) 2 and -3
  d) None of the above
  Answer: a
6. The discriminant of the equation
  2x2−4x+1=02x^2 – 4x + 1 = 0
  is:
  a) 4
  b) -4
  c) 0
  d) 8
  Answer: b
7. If the discriminant is negative, what type of roots does the quadratic equation have?
  a) Real and unequal
  b) Real and equal
  c) Imaginary
  d) None of the above
  Answer: c
8. The quadratic formula is:
  a)
  x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}
  b)
  x=b±b2+4ac2ax = \frac{b \pm \sqrt{b^2 + 4ac}}{2a}
  c)
  x=−b±b2+4acax = \frac{-b \pm \sqrt{b^2 + 4ac}}{a}
  d) None of the above
  Answer: a
9. A quadratic equation has at most how many real roots?
  a) 1
  b) 2
  c) 3
  d) Infinite
  Answer: b
10. The product of the roots of
  ax2+bx+c=0ax^2 + bx + c = 0
  is:
  a)
  −ba-\frac{b}{a}
  b)
  −ca-\frac{c}{a}
  c)
  ca\frac{c}{a}
  d)
  ba\frac{b}{a}
  Answer: c
11. The sum of the roots of
  ax2+bx+c=0ax^2 + bx + c = 0
  is:
  a)
  −ba-\frac{b}{a}
  b)
  −ca-\frac{c}{a}
  c)
  ba\frac{b}{a}
  d)
  ca\frac{c}{a}
  Answer: a
12. If the quadratic equation
  x2+4x+3=0x^2 + 4x + 3 = 0
  is solved by factorization, what are the roots?
  a) -1, -3
  b) 1, 3
  c) -1, 3
  d) None of the above
  Answer: a
13. If
  x=2x = 2
  and
  x=3x = 3
  are roots, what is the quadratic equation?
  a)
  x2−5x+6=0x^2 – 5x + 6 = 0
  b)
  x2+5x+6=0x^2 + 5x + 6 = 0
  c)
  x2−6x+5=0x^2 – 6x + 5 = 0
  d)
  x2+6x+5=0x^2 + 6x + 5 = 0
  Answer: a
14. Which of the following is the discriminant of the quadratic equation
  3x2−4x+5=03x^2 – 4x + 5 = 0
  ?
  a) -44
  b) 44
  c) 4
  d) -4
  Answer: a
15. If a quadratic equation has equal roots, the graph of the equation:
  a) Crosses the x-axis
  b) Does not intersect the x-axis
  c) Touches the x-axis at one point
  d) None of the above
  Answer: c
16. Solve
  2x2−3x+1=02x^2 – 3x + 1 = 0
  by factorization. The roots are:
  a) 1 and 1/2
  b) -1 and 1
  c) 1 and -1/2
  d) None of the above
  Answer: a
17. The vertex of the parabola represented by
  y=ax2+bx+cy = ax^2 + bx + c
  is given by:
  a)
  (−b/2a,c)(-b/2a, c)
  b)
  (−b/2a,f(−b/2a))(-b/2a, f(-b/2a))
  c)
  (b/2a,f(b/2a))(b/2a, f(b/2a))
  d) None of the above
  Answer: b
18. If
  a+b+c=0a + b + c = 0
  in
  ax2+bx+c=0ax^2 + bx + c = 0
  , one root of the quadratic equation is:
  a) 0
  b) 1
  c) -1
  d) None of the above
  Answer: c
19. The quadratic equation whose roots are -2 and 4 is:
  a)
  x2−2x−8=0x^2 – 2x – 8 = 0
  b)
  x2+2x−8=0x^2 + 2x – 8 = 0
  c)
  x2−2x+8=0x^2 – 2x + 8 = 0
  d)
  x2+2x+8=0x^2 + 2x + 8 = 0
  Answer: a
20. The quadratic equation representing the area of a rectangle with length
  2x+32x + 3
  and breadth
  x−1x – 1
  is:
  a)
  2x2+5x−3=02x^2 + 5x – 3 = 0
  b)
  2x2+5x−4=02x^2 + 5x – 4 = 0
  c)
  2x2+5x+4=02x^2 + 5x + 4 = 0
  d) None of the above
  Answer: b
21. If one root of the quadratic equation $x^2 – 7x + 12 = 0$ is 4, what is the other root?
  a) 2
  b) 3
  c) 8
  d) 5
  Answer: b
22. The graph of $y = x^2 – 4x + 4$ is:
  a) A straight line
  b) A parabola that touches the x-axis
  c) A parabola that does not intersect the x-axis
  d) None of the above
  Answer: b
23. The quadratic equation $2x^2 + 3x – 2 = 0$ is solved using the quadratic formula. What is the discriminant?
  a) 17
  b) -17
  c) 13
  d) -13
  Answer: a
24. What are the roots of the equation $x^2 – 5x + 6 = 0$ using factorization?
  a) 1 and 6
  b) -1 and -6
  c) 2 and 3
  d) -2 and -3
  Answer: c
25. If the product of two roots of the quadratic equation $ax^2 + bx + c = 0$ is 10, what is ?
  a)
  b)
  c)
  d) None of the above
  Answer: a
26. What is the nature of the roots if the discriminant is exactly zero?
  a) Real and unequal
  b) Real and equal
  c) Imaginary
  d) No roots
  Answer: b
27. If is a root of the quadratic equation $x^2 + bx – 2 = 0$ , what is $b$ ?
  a) 2
  b) -2
  c) 1
  d) -1
  Answer: c
28. A quadratic equation with no real roots will have:
  a) Positive discriminant
  b) Zero discriminant
  c) Negative discriminant
  d) None of the above
  Answer: c
29. Which of the following is not a quadratic equation?
  a) $3x^2 + 5x – 2 = 0$
  b)
  c) $2x^3 – x + 4 = 0$
  d) $4x^2 – x = 0$
  Answer: c
30. Solve $x^2 – 4x – 5 = 0$ using factorization. The roots are:
  a) -1 and 5
  b) 1 and -5
  c) -1 and -5
  d) 1 and 5
  Answer: a
31. If the vertex of $y = 3x^2 – 12x + 5$ is , what is the value of ?
  a) 3
  b) 2
  c) 1
  d) -2
  Answer: b
32. The sum of the roots of $x^2 – 6x + 9 = 0$ is:
  a) 6
  b) -6
  c) 3
  d) -3
  Answer: a
33. The quadratic equation whose roots are and 2 is:
  a) $2x^2 – 5x + 2 = 0$
  b) $2x^2 + 5x – 2 = 0$
  c) $x^2 – 5x + 4 = 0$
  d) None of the above
  Answer: a
34. What is the quadratic equation whose sum and product of roots are -3 and 2, respectively?
  a) $x^2 + 3x – 2 = 0$
  b) $x^2 – 3x + 2 = 0$
  c) $x^2 + 3x + 2 = 0$
  d) $x^2 – 3x – 2 = 0$
  Answer: a
35. If is a root of the quadratic equation $ax^2 + bx + c = 0$ , what is ?
  a) 0
  b)
  c)
  d) None of the above
  Answer: a
36. The quadratic equation $4x^2 + 12x + 9 = 0$ has roots:
  a) 3 and -3
  b) -3/2 and -3/2
  c) 3/2 and 3/2
  d) None of the above
  Answer: b
37. The equation $x^2 + px + 1 = 0$ has roots and . What is $p$ ?
  a) 3/2
  b) -5/2
  c) -3/2
  d) 5/2
  Answer: b
38. Solve $3x^2 – 5x – 2 = 0$ using the quadratic formula. The roots are:
  a) 2 and -1/3
  b) -2 and 1/3
  c) 1 and -1/2
  d) None of the above
  Answer: a
39. The equation $2x^2 + 8x + k = 0$ has real and equal roots. What is ?
  a) 8
  b) 16
  c) 4
  d) 64
  Answer: b
40. If the quadratic equation $x^2 – 4x + k = 0$ has equal roots, is:
  a) 4
  b) 0
  c) 2
  d) None of the above
  Answer: a
41. If one root of the quadratic equation $x^2 – px + q = 0$ is zero, what is ?
  a) 1
  b) 0
  c)
  d) None of the above
  Answer: b
42. The graph of $y = ax^2 + bx + c$ opens upward if:
  a)
  b)
  c)
  d) None of the above
  Answer: a
43. The roots of $x^2 + 6x + 8 = 0$ are:
  a) -2 and -4
  b) 2 and 4
  c) -2 and 4
  d) None of the above
  Answer: a
44. Which of the following quadratic equations has imaginary roots?
  a) $x^2 + 4x + 4 = 0$
  b) $x^2 + x + 1 = 0$
  c) $x^2 – 5x + 6 = 0$
  d) $x^2 – 4x + 3 = 0$
  Answer: b
45. The axis of symmetry of the parabola $y = ax^2 + bx + c$ is:
  a)
  b)
  c)
  d) None of the above
  Answer: a
46. If the quadratic equation $x^2 + px + q = 0$ has roots and , then is:
  a) 5
  b) -1
  c) -5
  d) 1
  Answer: c
47. The quadratic equation $3x^2 – 6x + k = 0$ has one root as 2. What is ?
  a) -2
  b) 0
  c) 6
  d) 3
  Answer: c
48. Solve $x^2 + 7x + 10 = 0$ . The roots are:
  a) -2 and -5
  b) 2 and 5
  c) -2 and 5
  d) None of the above
  Answer: a
49. The roots of $x^2 – 3x + 2 = 0$ are:
  a) 1 and 2
  b) -1 and -2
  c) -1 and 2
  d) 3 and 2
  Answer: a
50. What is the sum of the roots of $x^2 + 5x + 4 = 0$ ?
  a) -5
  b) 5
  c) -4
  d) 4
  Answer: a