Factor any trinomial ax² + bx + c into (px + q)(rx + s) step by step — handles the simple x² + bx + c case (a = 1) and the harder ax² + bx + c case (a ≠ 1) using the AC method with factor-pair grouping. Also finds roots and verifies with FOIL.
Find p, q such that p · q = c and p + q = b
When the leading coefficient is 1, factoring is a hunt for two numbers that multiply to c and add to b. Once found, the answer is simply (x + p)(x + q).
Example: x² + 5x + 6 — we need p·q = 6 and p+q = 5. Testing pairs: 1×6 = 6 but 1+6 = 7 ✗; 2×3 = 6 and 2+3 = 5 ✓. So the answer is (x + 2)(x + 3).
Negative c: one factor is positive and one is negative. Positive c, negative b: both factors are negative.
AC = a·c → split bx → factor by grouping
When a ≠ 1, find two numbers p and q such that p·q = a·c and p+q = b. Then rewrite the middle term as px + qx and factor by grouping in two pairs.
Example: 2x² + 7x + 3 — AC = 2·3 = 6, and we need p+q = 7. Since 1+6 = 7 and 1·6 = 6 ✓, rewrite as 2x² + x + 6x + 3, group as x(2x+1) + 3(2x+1), then factor out (2x+1) to get (x+3)(2x+1).
One-on-one Algebra 1 tutoring makes factoring click — we work through your actual problems and build pattern recognition that sticks for tests and beyond.