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Factoring Trinomials Calculator (ax² + bx + c) with Steps

Factor any trinomial ax² + bx + c over the integers with the ac (grouping) method, step by step: GCF, the two numbers, splitting the middle term and the final factored form.

a (x² coefficient)

b (x coefficient)

c (constant)

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How it works

ax² + bx + c: find m, n with m·n = a·c and m + n = b → ax² + mx + nx + c → group and factor → (px + q)(rx + s)

The ac method (factoring by grouping) works for any trinomial that factors over the integers. After pulling out the greatest common factor, multiply a by c and look for two integers whose product is ac and whose sum is b. Splitting the middle term bx into those two pieces lets you factor the first pair and the second pair separately; both groups then share the same binomial, which you factor out. If no such pair exists the trinomial is prime over the integers — the calculator then gives its real roots, which you can also get from the quadratic formula.

Worked example

Factor 6x² + 7x − 3. Here a × c = 6 × (−3) = −18, and the two numbers −2 and 9 multiply to −18 and add to 7. Split the middle term: 6x² − 2x + 9x − 3. Group: 2x(3x − 1) + 3(3x − 1). Factor out the common binomial: (3x − 1)(2x + 3). The roots are x = 1/3 and x = −3/2.

Sources: Factoring quadratics — Math is Fun · Factorization of polynomials — Wikipedia

Frequently asked questions

How do I know a trinomial cannot be factored?

If no pair of integers multiplies to ac and adds to b, it is prime over the integers. Equivalently, the discriminant b² − 4ac is not a perfect square.

Why factor out the GCF first?

It keeps the numbers small and guarantees the remaining factors are as simple as possible. 4x² + 8x − 12 becomes 4(x² + 2x − 3) = 4(x + 3)(x − 1).

What are the special patterns?

A perfect square trinomial a² ± 2ab + b² factors as (a ± b)², and a difference of squares a² − b² factors as (a + b)(a − b). The calculator labels these when it finds them.

Is the order of the factors important?

No. (3x − 1)(2x + 3) and (2x + 3)(3x − 1) are the same product. Expanding either one with FOIL gives back the original trinomial.

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This calculator is for educational purposes. Double-check important results before acting on them.