How factoring works
You are looking for two binomials that multiply back to the original trinomial. For 2x² + 7x + 3 they are (x + 3) and (2x + 1).
The usual method is to find two numbers that multiply to a × c and add to b, split the middle term, then group.
Worked example
For 2x² + 7x + 3, a × c is 6, and the two numbers that multiply to 6 and add to 7 are 6 and 1.
Split 7x into 6x + x, group as 2x(x + 3) + 1(x + 3), and the answer is (x + 3)(2x + 1).
Its discriminant is 25, a perfect square, which is why whole-number binomials exist.
Greatest common factor first
Check whether one number divides all three coefficients before you start. 6x² + 9x + 3 has a greatest common factor of 3, so it becomes 3(2x² + 3x + 1).
Leaving it out is the most common mistake, because the binomials alone do not multiply back to the original.
Factoring and standard form
American textbooks say factor and call the result factored form. British textbooks say factorise and factorised form.
The arithmetic is identical, so an answer written one way converts to the other without changing the numbers.
Inputs for the factoring calculator
Coefficient of x²
The number multiplying x². It cannot be zero.
Coefficient of x
The number multiplying x, including its sign.
Constant
The number on its own. Whole numbers only, because the binomials are searched for exactly.
CALCULATOR QUESTIONS
Factoring calculator questions
How do I factor 2x² + 7x + 3?
Find two numbers multiplying to 6 and adding to 7, which are 6 and 1. Splitting and grouping gives (x + 3)(2x + 1).
Is factoring the same as factorising?
Yes. US courses say factor and UK schools say factorise for the same process.
Why pull out the greatest common factor first?
The answer is only complete once it is outside the parentheses. 6x² + 9x + 3 is 3(2x + 1)(x + 1).
What if it will not factor?
Then the roots are irrational, so no whole-number binomials exist. Use the quadratic formula instead.