A set of practice questions, with answers, on quadratic equations are presented. The answers are at the bottom of the page, and full solutions with explanations are also available.
What are the two solutions to the quadratic equation:
\[ 2x^2 + 3x - 2 = 0 \]
A) \(-2 , 3\)
B) \(-2 , -\frac{1}{2}\)
C) \(2 , -\frac{1}{2}\)
D) \(-2 , \frac{1}{2}\)
E) \(-\frac{1}{2} , -2\)
What is the sum of the two solutions to the quadratic equation:
\[ (x + 4)(x - 3) = 7 \]
A) \(-1\)
B) \(-2\)
C) \(1\)
D) \(2\)
E) \(3\)
What is the product of the two solutions to the quadratic equation:
\[ (x - 2)(x - 6) = -3 \]
A) \(12\)
B) \(-12\)
C) \(15\)
D) \(-3\)
E) \(3\)
Find all values of \(m\) for which the quadratic equation has no real solutions:
\[ x^2 + 2x - 2m = 0 \]
A) \(m = 0\)
B) \(m = -2\)
C) \(m < -4\)
D) \(m = \frac{1}{2}\)
E) \(m < -\frac{1}{2}\)
Find all values of \(m\) for which the quadratic equation has two distinct real solutions:
\[ 2x^2 + 3x - m + 2 = 0 \]
A) \(m > 0\)
B) \(m = 7\)
C) \(m > \frac{7}{8}\)
D) \(m < -\frac{7}{8}\)
E) \(m = \frac{7}{8}\)
Which of these quadratic equations has two real solutions greater than zero?
A) \(x^2 + x = 0\)
B) \(2x^2 - 10x = 28\)
C) \(-x^2 + 4x + 5 = 0\)
D) \(-3x^2 - 9 = -12x\)
E) \(-3x^2 - 6x + 24 = 0\)
Which of these quadratic equations has two real solutions whose product is greater than zero?
A) \(-x^2 - 2x = -8\)
B) \(x^2 + 9x = -18\)
C) \(-x^2 = -6 + x\)
D) \(x^2 = 4x\)
E) \(x^2 - 3x = 4\)
\(b\) and \(c\) in the quadratic equation \(x^2 + bx + c = 0\) are real numbers. Find \(b\) and \(c\) so that the given equation has two solutions \(x = -\frac{1}{4}\) and \(x = \frac{1}{2}\).
A) \(b = -\frac{1}{4}, c = -\frac{1}{8}\)
B) \(b = -1, c = -1\)
C) \(b = \frac{1}{4}, c = -1\)
D) \(b = \frac{1}{4}, c = \frac{1}{8}\)
E) \(b = 4, c = 8\)
\(b\) and \(c\) in the quadratic equation \(-x^2 + bx + c = 0\) are real numbers. Find \(b\) and \(c\) so that the given equation has two solutions whose sum is 6 and whose product is 8.
A) \(b = 6, c = 8\)
B) \(b = -6, c = 8\)
C) \(b = 8, c = -6\)
D) \(b = 6, c = -8\)
E) \(b = -8, c = 6\)
Which of these pairs of quadratic equations have the same solutions (equivalent equations)?
A) \(x^2 - 1 = 0\) and \(x^2 = -1\)
B) \(-x^2 + x = -6\) and \(x^2 - 2x = 3\)
C) \(x^2 - 5x + 6 = 0\) and \(-x^2 - 5x - 6 = 0\)
D) \(x^2 = 2x\) and \(x^2 + 2x = 0\)
E) \(x^2 + x - 2 = 0\) and \(-2x^2 - 2x + 4 = 0\)