Factoring Calculator

What this calculator factors

The supported input is ax^2 + bx + c, with integer values for a, b and c. This covers the most common classroom factoring problems: monic quadratics, leading-coefficient quadratics, expressions with a shared GCF, and linear expressions when a = 0.

The standard strategy is:

1. Factor out the integer GCF from all terms.
2. On the remaining quadratic, compute a*c.
3. Find two integers m and n such that m*n = a*c and m+n = b.
4. Rewrite bx as mx + nx, group the four terms, and factor the shared binomial.

Worked example

Factor 6x^2 + 11x - 10.

This is different from a quadratic equation solver. A solver focuses on values of x that make the expression equal zero. This tool focuses on rewriting the expression itself as a product of factors. The roots are included only as a check: each linear factor gives one root.

When no integer factorization exists

Some quadratics do not factor cleanly over the integers. For example, x^2 + x + 1 has discriminant -3, so it has no real linear factors. The expression x^2 - 2 has real roots but no integer factor pair, so it is not factorable over the integers even though it can be written with irrational factors.

Common pitfalls

• Skipping the GCF. 2x^2 + 10x + 12 should become 2(x^2 + 5x + 6) before the pair search.
• Using c instead of a*c. When a is not 1, the product target is a*c, not only c.
• Losing signs. A negative constant means one number in the pair is positive and the other is negative.
• Confusing factoring with roots. Factored form and roots are connected, but they answer different questions.