How to Solve Quadratic Equations: 3 Methods Explained

A quadratic equation has the form ax² + bx + c = 0. There are three main methods to solve it, each useful in different situations.

Method 1: Factoring

If the quadratic can be factored, this is the fastest method. Find two numbers that multiply to a×c and add to b. Example: x² + 7x + 12 = 0. Find two numbers that multiply to 12 and add to 7: 3 and 4. So (x+3)(x+4) = 0, giving x = −3 or x = −4.

Method 2: The Quadratic Formula

Works for any quadratic equation. x = (−b ± √(b² − 4ac)) / 2a. Example: 2x² − 4x − 6 = 0. a=2, b=−4, c=−6. x = (4 ± √(16+48)) / 4 = (4 ± 8) / 4. x = 3 or x = −1. Use this when factoring is difficult or impossible.

Method 3: Completing the Square

Rewrite the equation as (x + d)² = e. Example: x² + 6x − 7 = 0. Move the constant: x² + 6x = 7. Add (6/2)² = 9 to both sides: x² + 6x + 9 = 16. Factor: (x+3)² = 16. Take square root: x+3 = ±4. x = 1 or x = −7.

Which method should I use?

Try factoring first — it is fastest when it works. If you cannot find factors quickly, use the quadratic formula. Completing the square is useful for deriving the quadratic formula itself and for graphing parabolas in vertex form.

The Discriminant

The value b² − 4ac (inside the square root) tells you the nature of the roots. If positive: two real roots. If zero: one real root (repeated). If negative: two complex roots. Check this before solving to know what to expect.

Practice solving equations with the Math Champion algebra quiz, or review the quadratic formula reference page.