Quadratic Formula and Discriminant Practice Problems

How it works

For ax² + bx + c = 0 (with a ≠ 0),

x = (−b ± √(b² − 4ac)) ÷ 2a

The expression under the square root, D = b² − 4ac, is the discriminant:

Before using the formula, divide out any common factor and try factoring. It is often faster.

These problems come from Mathyessy's question bank. Try each one before opening the solution.

What is the discriminant of x²+3x+5 = 0?
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Answer: −11
b² − 4ac = 3² − 4 × 1 × 5 = 9 − 20 = −11. It is negative, so there are no real roots.
What is the discriminant of x²+9x+2 = 0?
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Answer: 73
9² − 4 × 1 × 2 = 81 − 8 = 73. That is positive but not a perfect square, so there are two irrational real roots.
x²+16x+64 = 0 has a repeated root. What is the positive value of k with x²+kx+64 = 0 also repeated?
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Answer: 16
A repeated root needs D = 0: k² − 4 × 64 = 0, so k² = 256 and the positive value is k = 16. (x² + 16x + 64 = (x + 8)².)
15x²+48x+36 = 0. What is the larger root?
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Answer: −6/5
Divide by 3: 5x² + 16x + 12 = 0. This factors as (5x + 6)(x + 2) = 0, so the roots are −6/5 and −2. The larger is −6/5 = −1.2.

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