How to Solve a Quadratic Equation

Method One: Factorising

Rearrange into ax² + bx + c = 0, then look for two numbers that multiply to ac and add to b. For x² − 7x + 10 those are −5 and −2, giving (x − 5)(x − 2) = 0 and roots of 5 and 2. This is fastest when the coefficients are integers with a whole number answer, and it confirms the answer as you go.

Method Two: Completing the Square

Rewrite as x² + (b/a)x = −c/a, then add (b/2a)² to both sides so the left becomes a perfect square. Take square roots of both sides and isolate x. This is the derivation of the quadratic formula and is worth learning once, but it is far too long to use routinely.

Method Three: The Quadratic Formula

Substitute into x = (−b ± √(b² − 4ac)) / 2a. This works for every quadratic with a non zero a, including ones that cannot be factored over the rationals. Its weakness is not correctness but arithmetic, since it is easy to drop the negative in front of b or the denominator of 2a.

How to Choose

If a = 1 and the discriminant is a perfect square, factorise. If you need to derive the formula or the numbers are messy, complete the square. For everything else, and especially under time pressure, use the formula and verify with Vieta's relationships.

The Non Zero a Requirement

All three methods assume a is not zero. With a = 0 the equation is linear, solved as x = −c / b, and the quadratic formula is undefined because it divides by a. Check this before doing anything else.

Always Verify

The roots of ax² + bx + c = 0 must sum to −b / a and multiply to c / a. Checking both takes seconds and catches sign errors immediately. If your answer fails either check, the most likely cause is a sign that was lost moving terms across the equals sign.