Factoring Calculator

What Factoring Means and How to Do It

Factoring a quadratic expression means rewriting it as a product of two simpler expressions, usually two binomials, instead of a sum of terms. If expanding (x + 2)(x + 3) gives you x² + 5x + 6, factoring reverses that process: starting from x² + 5x + 6, you work backward to recover (x + 2)(x + 3). This matters because factored form makes it easy to find where an expression equals zero: a product is zero only when one of its factors is zero, so factoring is often the fastest route to solving a quadratic equation.

Several standard methods handle different situations. When every term shares a common factor, you pull out the greatest common factor first, such as rewriting 2x² + 4x as 2x(x + 2). For trinomials in the form ax² + bx + c, the most reliable general technique is the ac method, also called factoring by grouping: find two numbers that multiply to a times c and add to b, use them to split the middle term into two pieces, then group and factor each pair separately until a common binomial emerges. Special patterns are worth recognizing on sight too, including a difference of squares, a² − b², which always factors as (a + b)(a − b), and perfect square trinomials like a² + 2ab + b².

Not every quadratic factors neatly into integers. The calculator above checks this automatically using the discriminant, b² − 4ac: when it is a positive perfect square, integer factors exist and the ac method finds them step by step. When the discriminant is zero, the expression is a perfect square trinomial with a single repeated factor. When it is positive but not a perfect square, the roots are irrational, and when it is negative, there are no real factors at all, only complex ones. Recognizing which case applies before you start saves time and avoids forcing a factorization that does not exist.