Solving a Quadratic by Factoring

Solving a quadratic by factoring means writing standard form, factoring the quadratic side completely, and applying the zero-product property to each factor.

Solve x² − x − 6 = 0. Standard form holds already. The pair search wants two numbers multiplying to −6 and adding to −1: the pair is −3 and 2, so the factoring is (x − 3)(x + 2) = 0. The zero-product property splits it: x − 3 = 0 or x + 2 = 0, giving x = 3 or x = −2. Check both in the original: 9 − 3 − 6 = 0 and 4 + 2 − 6 = 0 — two candidates, two survivors. A quadratic can hold two solutions, and factoring hands over both at once.