Solving Quadratics by Factoring

The whole method rests on one fact: if two things multiply to zero, at least one of them is zero.

To solve x² + 5x + 6 = 0:
Factor: (x + 2)(x + 3) = 0
Set each factor to zero: x = −2 or x = −3.

The equation must equal zero first. For x² + 5x = −6, move everything to one side before factoring. Factoring while the other side is 6 tells you nothing, because two numbers can multiply to 6 in infinitely many ways.

Solutions of a quadratic are also called roots or zeros, and they are exactly the x-intercepts of the parabola.

The zero product property holds because ℝ has no zero divisors: ab = 0 ⟹ a = 0 or b = 0. No analogous property exists for any other constant, which is precisely why the equation must be arranged in the form (expression) = 0 before factoring.

Factoring solves only those quadratics whose roots are rational, equivalently those whose discriminant b² − 4ac is a perfect square. When it is not, the roots are irrational or complex and another method is required — which is why factoring is fast but not general.

A repeated factor gives a double root: (x − 3)² = 0 has the single solution x = 3 with multiplicity 2, and the parabola touches the x-axis there rather than crossing it.

Also note that dividing both sides by a variable factor destroys solutions. Solving x² = 5x by dividing by x loses x = 0; factoring to x(x − 5) = 0 keeps both.