Solving Quadratics by Factoring

Set it to zero, factor, and use the zero-product property.

The idea

To solve a quadratic equation is to find the values of $x$ that make it equal $0$. Those values are its roots, and on a graph they are the places where the curve crosses the $x$-axis.

Inverse operations cannot isolate $x$ here, because a quadratic contains $x$ in two places at once. Instead we factor: rewrite the quadratic as a product. A product equal to $0$ is valuable because of the zero-product property: $AB = 0 \text{ if and only if } A = 0 \text{ or } B = 0,$ which holds because two nonzero numbers always multiply to something nonzero. No other target value behaves this way — $AB = 6$ tells us nothing about $A$ on its own — which is why one side must be $0$ before factoring helps.

The procedure has three steps. Arrange the equation so that one side is $0$. Factor the other side. Then set each factor equal to $0$ in turn and solve the short equations that result; every factor contributes a root. A quadratic with roots $r_1$ and $r_2$ factors as $(x - r_1)(x - r_2)$, and its graph crosses the $x$-axis at exactly those two points.

Ways to work on it

• Walkthrough. Zero-product property on two factorable quadratics.
• Practice. Factor a quadratic with leading coefficient 1 and read off both roots.
• Hardest. A leading coefficient bigger than 1 — split the middle term and factor.

Not sure where to start? Take the ten-question placement test.