Enter the coefficients of your quadratic trinomial (ax² + bx + c) and the Factoring Trinomials Calculator returns the fully factored form, the two roots, and a step-by-step breakdown using the AC method. Works for trinomials with any leading coefficient — just fill in a, b, and c.
Factored Form
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Root 1 (x₁)
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Root 2 (x₂)
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Discriminant (b² − 4ac)
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Result Type
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A trinomial is a polynomial with three terms, typically written as ax² + bx + c. Factoring means rewriting it as a product of two binomials, like (px + q)(rx + s). The most common method is the AC method: multiply a × c, find two numbers that multiply to ac and add to b, then split the middle term and factor by grouping.
In the AC method, you multiply the leading coefficient a by the constant c to get the product AC. Then you find two integers p and q such that p × q = AC and p + q = b. You rewrite the middle term bx as px + qx, then factor the resulting four-term polynomial by grouping into two binomials.
If the discriminant (b² − 4ac) is negative, the trinomial has no real roots and cannot be factored over the real numbers. In that case it is called irreducible over ℝ, though it can be factored using complex numbers. This calculator will report that the trinomial is not factorable over the reals.
Yes. By entering any integer value for the coefficient a (the leading coefficient of x²), the calculator handles non-monic trinomials like 2x² + 7x + 3 or 6x² − 5x − 6 using the AC method. Just fill in all three fields: a, b, and c.
Before using the AC method, you should factor out the GCF from all three coefficients. For example, 2x² + 4x + 2 should first be simplified to 2(x² + 2x + 1), then the inner trinomial factored. This calculator works directly on the coefficients you enter, so divide out any GCF yourself first for the cleanest result.
Expand the two binomials by multiplying them out using FOIL (First, Outer, Inner, Last) and verify that you get back the original trinomial ax² + bx + c. Alternatively, substitute the roots into the original expression — if both roots make the expression equal zero, the factoring is correct.
A perfect square trinomial factors into (px + q)², meaning both binomials are identical. This happens when the discriminant equals zero, i.e., b² = 4ac. Examples include x² + 6x + 9 = (x + 3)² and 4x² − 12x + 9 = (2x − 3)². The calculator will identify this special case.
Roots (also called zeros or solutions) are the values of x that make the trinomial equal to zero. Factors are the binomial expressions whose product equals the trinomial. If the roots are x₁ and x₂, the factored form is a(x − x₁)(x − x₂). Both convey the same information in different forms.