Factoring quadratics

Factoring a quadratic means rewriting an expression of the form ax² + bx + c as a product of simpler factors — usually two binomials. Since multiplying binomials uses FOIL (First, Outer, Inner, Last), factoring is often called reverse FOIL: you're undoing the multiplication to recover the original factors.

For a quadratic with a = 1, the method is to find two numbers that multiply to c and add to b. To factor x² + 5x + 6, look for a pair multiplying to 6 and summing to 5: that's 2 and 3, so x² + 5x + 6 = (x + 2)(x + 3). Other standard tools include pulling out a greatest common factor first (2x² + 8x = 2x(x + 4)), recognizing a difference of squares (x² − 9 = (x + 3)(x − 3)), and spotting perfect square trinomials (x² + 6x + 9 = (x + 3)²). When a ≠ 1, grouping or trial-and-error with factor pairs of a and c does the job.

Factoring matters because of the zero-product property: if a product equals zero, at least one factor must be zero. So once x² + 5x + 6 = 0 is factored to (x + 2)(x + 3) = 0, the solutions are x = −2 and x = −3. Those roots are also the parabola's x-intercepts, linking factored form directly to graphs. Quadratics that don't factor neatly can be solved by completing the square or the quadratic formula instead.

Factoring quadratics is tested constantly on the ACT, SAT, and CLT — from straightforward "solve by factoring" problems to questions connecting factored form with intercepts, and factoring rational expressions to simplify them. Fast, accurate factoring is one of the highest-value algebra skills for all three exams.