The Quadratic Formula: Why It Works and When to Use It

The quadratic formula solves ax^2 + bx + c = 0 for any real coefficients a, b, c where a ≠ 0. Here’s the intuition and the practical checklist for using it.

The Formula

x = [-b ± sqrt(b^2 - 4ac)] / (2a)

When to Use It

• Factoring is messy or not obvious.
• You need exact roots with radicals.
• You want to analyze the discriminant quickly.

Interpreting the Discriminant

If b^2 - 4ac > 0, you get two distinct real roots; if it equals 0, one repeated real root; if < 0, two complex roots.

Related Reading

Prefer a deeper, example-driven explanation? See our full page: Quadratic Formula: Step-by-Step.