Factoring Calculator

How it works

It takes out any common factor, then looks for two integers that multiply to a·c and add to b, and factors by grouping. When no such integers exist, the trinomial doesn’t factor over the integers, and the quadratic formula is the way to find its roots.

Example

Factor 6x² + 7x − 3.

(3x − 1)(2x + 3)

Steps

1. 1
   Find two integers that multiply to a·c = −18 and add to b = 7: 9 and −2.
2. 2
   Split the middle term: 6x² + 9x − 2x − 3.
3. 3
   Group the pairs: 3x(2x + 3) − 1(2x + 3) = (3x − 1)(2x + 3).
4. 4
   So 6x² + 7x − 3 = (3x − 1)(2x + 3).

More examples

Factor x² − 9x + 20.

(x − 5)(x − 4)

Steps

1. 1
   Find two integers that multiply to a·c = 20 and add to b = −9: −4 and −5.
2. 2
   Split the middle term: x² − 4x − 5x + 20.
3. 3
   Group the pairs: x(x − 4) − 5(x − 4) = (x − 5)(x − 4).
4. 4
   So x² − 9x + 20 = (x − 5)(x − 4).

Factor 4x² − 25.

(2x − 5)(2x + 5)

Steps

1. 1
   Find two integers that multiply to a·c = −100 and add to b = 0: 10 and −10.
2. 2
   Split the middle term: 4x² + 10x − 10x − 25.
3. 3
   Group the pairs: 2x(2x + 5) − 5(2x + 5) = (2x − 5)(2x + 5).
4. 4
   So 4x² − 25 = (2x − 5)(2x + 5).

Common mistake

Signs are where this goes wrong. If c is negative, the two numbers have opposite signs. If c is positive, they share the sign of b.

Questions

How do I check my answer?

Multiply the factors back out. (2x + 3)(3x − 1) = 6x² + 7x − 3, which matches.

What if a is negative?

Take out −1 first so the x² term is positive, then factor what is left.

Does every trinomial factor?

No. For x² + x + 1, no two integers multiply to 1 and add to 1, so it does not factor over the integers.

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