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The Quadratic Formula and Discriminant

Algebra Foundation

Algebra 1

Study expressions, equations, inequalities, functions, linear models, systems, polynomials, quadratics, and exponential growth with original explanations and worked examples.

Learn with original explanations and worked examples. OpenStax Elementary Algebra 2e was consulted for topic coverage.

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Algebra 1 · Unit 9 · Topic 9.4

The Quadratic Formula and Discriminant

For ax² + bx + c = 0 with a ≠ 0, the solutions are x = [−b ± √(b² − 4ac)]/(2a).

Definition and explanation

For ax² + bx + c = 0 with a ≠ 0, the solutions are x = [−b ± √(b² − 4ac)]/(2a). The discriminant D = b² − 4ac gives two distinct real roots if D > 0, one repeated real root if D = 0, and no real roots if D < 0.

Worked example

Solve 2x² − 3x − 2 = 0. Here a = 2, b = −3, and c = −2, so D = 9 + 16 = 25. Thus x = (3 ± 5)/4, giving x = 2 or −1/2; substitution yields zero for both.

Common mistake

The denominator 2a divides the entire numerator; keep the sign of b when calculating −b.

Key takeaway

For ax² + bx + c = 0 with a ≠ 0, the solutions are x = [−b ± √(b² − 4ac)]/(2a).

Explain each step of the worked example, then check the units, signs, and conditions that apply.

Further reading

Topic coverage was checked against the openly available OpenStax Elementary Algebra 2e and Algebra and Trigonometry 2e textbooks. The explanations and examples on this site are independently written; textbook prose, exercises, and artwork are not reproduced.