Factoring Calculator

Factoring turns a quadratic like x² + 5x + 6 into (x + 2)(x + 3), which immediately reveals the roots. Enter the three coefficients and this calculator finds the integer factorization when one exists — and always reports the discriminant and the exact roots from the quadratic formula.

Results

Formula

To factor ax² + bx + c, find two numbers that multiply to a × c and add to b, then split the middle term and group. The roots always satisfy:
x = (−b ± √(b² − 4ac)) ÷ 2a

How to use this calculator

  1. Enter the coefficients a, b and c of your quadratic.
  2. Press Calculate.
  3. Read the factored form; if no integer factorization exists, use the roots from the quadratic formula.
Factoring Calculator — how the calculation works, illustrated
Factoring Calculator — how the calculation works, illustrated

Worked example

Factor x² + 5x + 6.

  1. Find two numbers multiplying to 6 and adding to 5: 2 and 3.
  2. Write the binomials: (x + 2)(x + 3).
  3. Set each to zero: x = −2 and x = −3.

Result: x² + 5x + 6 = (x + 2)(x + 3), with roots −2 and −3.

Frequently asked questions

What if the quadratic does not factor?

Many quadratics have no integer factorization. The roots still exist (possibly irrational or complex) and are given exactly by the quadratic formula, which this calculator also reports.

What does the discriminant tell me?

b² − 4ac: positive means two real roots, zero means one repeated root, negative means two complex roots.

How can I check a factored answer?

Multiply the binomials back out (FOIL). You should recover the original quadratic exactly.

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