Factoring calculator that shows which method applies
Enter a quadratic and get its factored form along with the reason that form was reached, be it a common factor, a difference of two squares, a perfect square or the ac method. When a quadratic genuinely doesn't factor over the integers, it says so and explains why rather than returning the expression unchanged.
Part of SolveStep's algebra solver. To solve rather than factor, use the quadratic equation solver.
Four patterns, checked in order
Factoring is pattern recognition more than calculation. The order matters: always take out a common factor first, because doing so often turns an awkward-looking quadratic into one of the easy patterns. Then check whether what remains is a difference of two squares or a perfect square, and only then reach for the general method.
The honest part most calculators skip: plenty of quadratics simply do not factor with whole numbers. x² + x − 1 has perfectly good roots, they are just irrational. Returning the expression unchanged, as many tools do, leaves you unsure whether the tool failed or the factorisation doesn't exist. Here the discriminant is shown and the situation is named.
Common factor first
6x² − 9x − 15 → 3(2x − 5)(x + 1)
The GCF of the coefficients comes out before anything else, which usually makes the rest straightforward.
Difference of two squares
x² − 9 → (x + 3)(x − 3)
No middle term and a negative constant, with both parts perfect squares. Worth recognising on sight.
Perfect square trinomial
x² − 6x + 9 → (x − 3)²
Both factors are identical, so it is written as a square rather than the same bracket twice.
Doesn't factor
x² + x − 1
The discriminant is 5, which is not a perfect square, so the roots are irrational and no integer factorisation exists. The tool says exactly that, and why.
Two quadratics, factored
The ac method
2x² + 7x + 3- There is no common factor, so go straight to the general method.
- Multiply a × c = 2 × 3 = 6. Find two numbers that multiply to 6 and add to b = 7: that is 1 and 6.
- Split the middle term using them: 2x² + 1x + 6x + 3.
- Group and factor each pair: x(2x + 1) + 3(2x + 1).
- The bracket (2x + 1) is now common to both.
Common factor hiding a perfect square
3x² + 6x + 3- All three coefficients share a factor of 3. Take it out: 3(x² + 2x + 1).
- What remains is a perfect square trinomial: the constant is 1² and the middle term is 2 × 1 × x.
- So x² + 2x + 1 = (x + 1)².
- Had the 3 not been removed first, the pattern would have been much harder to spot.
Factoring calculator FAQ
What does it mean when a quadratic won't factor?
It means there is no way to write it as brackets with whole numbers. Check the discriminant b² − 4ac: if it is not a perfect square the roots are irrational, and if it is negative there are no real roots at all. Both are stated explicitly rather than the expression being handed back unchanged.
Why take out the common factor first?
Because it usually reveals a pattern that was hidden. 3x² + 6x + 3 looks unremarkable until you remove the 3 and see (x + 1)². Factoring in the wrong order makes easy problems look hard.
What is the ac method?
Multiply a by c, find two numbers that multiply to that product and add to b, then use them to split the middle term into two so the expression can be grouped into pairs. It works for any factorable quadratic, including ones where a is not 1.
Does it factor cubics or higher powers?
Not yet. This handles quadratics, where the pattern rules are reliable. Higher-degree factoring usually needs the factor theorem and trial roots, which is a different procedure.
Is factoring the same as solving?
Closely related but not identical. Factoring rewrites the expression as a product; solving finds the x values that make it zero. Once factored, the roots follow immediately by setting each bracket to zero, which is the job of the quadratic equation solver.
Can I enter it as an equation?
Yes. Both x^2 - 5x + 6 and x^2 - 5x + 6 = 0 are accepted; the factored form is the same either way.