How to Factor a Trinomial with the AC Method
A trinomial is a three-term polynomial of the form ax² + bx + c, where a, b, and c are constants and a ≠ 0. Factoring rewrites it as a product of two binomials, such as (px + q)(rx + s), so that multiplying the binomials back out returns the original trinomial. This calculator uses the AC method (also called the product–sum method, or factoring by grouping) together with the discriminant and the quadratic formula, so it can factor any trinomial with real coefficients — whether the leading coefficient a is 1 or not, and even when the trinomial only factors over the reals or the complex numbers.
Step by step: the AC (product–sum) method
To factor ax² + bx + c, first multiply a and c together. Then look for two numbers, m and n, whose product equals a·c and whose sum equals b. Rewrite the middle term bx as mx + nx, splitting the trinomial into four terms: ax² + mx + nx + c. Group the first two terms and the last two terms, factor the greatest common factor out of each group, and — when m and n were chosen correctly — both groups share the same binomial factor, which then factors out to leave two binomials. For example, to factor 2x² + 7x + 3: a·c = 6, and the pair (1, 6) sums to 7, so 2x² + x + 6x + 3 = x(2x + 1) + 3(2x + 1) = (2x + 1)(x + 3).
When a trinomial doesn't factor over the integers
Not every trinomial breaks into binomials with whole-number coefficients. The discriminant D = b² − 4ac tells you what to expect before you start testing factor pairs. If D is a perfect square, integer (or rational) factors exist and the AC method will find them. If D is positive but not a perfect square, the roots x = (−b ± √D) / (2a) are irrational, and the trinomial only factors using those irrational roots, as a(x − x₁)(x − x₂). If D is negative, the trinomial has no real roots at all — it does not factor into real binomials, though it does factor over the complex numbers using a complex-conjugate pair of roots.
Checking a factored answer
Always verify a factorization by multiplying the binomials back out (FOIL) and confirming you recover the original ax² + bx + c. A faster check compares just the constant term — the product of the two second terms in the binomials — against c, and the cross-sum of the terms against b. This calculator performs both checks automatically and reports the discriminant and both roots alongside the factored form, so you can see exactly why an answer works.