Factoring Calculator
Factor a quadratic ax² + bx + c over the integers: pull out a common factor, then write a product of binomials when the AC method finds an integer pair.
- Write the polynomial: x² + 5x + 6.
- Find an integer pair that multiplies to a·c = 6 and adds to b = 5: 2 and 3.
- Write the product (x + 2)(x + 3) and multiply back to check.
- Factored form: (x + 2)(x + 3).
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All calculations run locally in your browser. Your entered values are not sent to an external service.
How to Use the Factoring Calculator
Enter integer coefficients a, b, and c for the trinomial ax² + bx + c, then select Factor. The tool pulls out a greatest common factor and, when an integer factorization exists, writes a product of binomials.
- Enter a, b, and c. Use 0 for a missing term, as in x² − 9.
- Select Factor, or pick a preset for a difference of squares, a perfect square, or an irreducible quadratic.
- Read the factored form and the substitution steps. Reset restores x² + 5x + 6.
This page factors over the integers. It does not factor cubics, and it does not invent real-number factors such as (x − √2). Use the Quadratic Formula Calculator when the discriminant is not a perfect square.
The AC Method
After you factor out a common integer, the remaining quadratic A x² + B x + C factors as (d x + e)(f x + g) when d f = A, e g = C, and d g + e f = B. The pair (e, B − e) is the AC pair: those two numbers multiply to A C and add to B.
A difference of squares is the special case B = 0 and C negative, as in x² − 9 = (x + 3)(x − 3). A perfect square has a repeated binomial, as in x² + 6x + 9 = (x + 3)².
Factoring Calculator Examples
A monic trinomial
x² + 5x + 6: a·c = 6 and b = 5, so the pair is 2 and 3. The product is (x + 2)(x + 3).
A difference of squares
x² − 9 = (x + 3)(x − 3). There is no x term, and both 9 and 1 are squares.
A leading coefficient other than 1
2x² + 7x + 3: a·c = 6 and b = 7, so the pair is 1 and 6. Grouping gives (2x + 1)(x + 3).
A perfect square
x² + 6x + 9 = (x + 3)². Expanding (x + 3)(x + 3) recovers the original trinomial.
An irreducible quadratic
x² + x + 1 has discriminant 1 − 4 = −3. No integer pair works. The page leaves the quadratic as written and sends you to the quadratic formula for the complex roots.
A common factor and an x
2x² + 4x = 2x(x + 2). First pull out 2, then factor x from x² + 2x.
How This Differs from the Quadratic Formula
Factoring is the method you use when integer binomials exist. The quadratic formula works whether the roots are integers, fractions, irrationals, or complex. If this calculator reports that the quadratic does not factor over the integers, the roots still exist — they just are not integer (or simple rational) intercepts you would write as (x − r) with integer r.
The Equation Solver on SolverCalc handles linear equations in x. The GCD Calculator is the number-theory tool behind the common-factor step.
Common Factoring Mistakes
Forgetting the greatest common factor
2x² + 10x + 12 is 2(x² + 5x + 6) = 2(x + 2)(x + 3), not only (2x + 4)(x + 3) without simplifying.
Signing the pair incorrectly
For x² − 5x + 6 the pair is −2 and −3, because the product is positive and the sum is negative.
Treating every quadratic as factorable
x² + 1 and x² + x + 1 do not factor over the integers. That is a complete answer, not a failure.
Writing (x + 3)(x + 3) instead of (x + 3)²
Both are correct. The squared form is the usual classroom name for a perfect square.
Entering decimals
This page requires integers. Clear a decimal equation of fractions first, or use the quadratic formula on the original coefficients.
Factoring Calculator FAQs
What does this calculator factor?
Integer quadratics ax² + bx + c, including linear expressions when a = 0 and differences of squares.
What is the AC method?
Find two integers that multiply to a·c and add to b, then split the middle term and group.
What if it does not factor?
The tool says the quadratic is irreducible over the integers. Use the Quadratic Formula Calculator for the roots.
Can I enter a negative leading coefficient?
Yes. The calculator factors out −1 so the remaining quadratic has a positive x² coefficient.
Does this factor cubics or expressions with two variables?
No. One quadratic in x only.
Are my numbers sent to a server?
No. The calculation runs locally in your browser.
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Reviewed by Prof. Dr. Khalil Mudassar (PhD) · Last updated: [mc_modified_date]
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Disclaimer
This calculator is for general educational use. It factors integer quadratics only. Review important results before relying on them in professional work.