What if the quadratic cannot be factored nicely with integers?

The calculator still uses the quadratic formula to find the roots and can express the answer using fractions or square roots. It also tells you when the quadratic is irreducible over the real numbers, meaning any factorization would require complex numbers.

All calculators

Factoring Calculator

Q: What does the Factoring Calculator do?

The Factoring Calculator takes a quadratic of the form ax² + bx + c, computes the discriminant and roots, and then writes the expression in factored form when possible. It also explains each step so you can see how the factorization works.

Q: When can a quadratic be factored over the real numbers?

A quadratic can be factored over the real numbers when its discriminant D = b² − 4ac is greater than or equal to zero. If D > 0, there are two distinct real roots; if D = 0, there is one repeated root; if D < 0, the quadratic has no real roots and is irreducible over the reals.

Factor quadratic polynomials of the form ax² + bx + c. Enter the coefficients, and we will compute the discriminant, roots, and a factored form (when possible), with clear step-by-step explanations and a small graph of the parabola.

Background

Many quadratics can be written in factored form a(x - r₁)(x - r₂), where r₁ and r₂ are the roots. These are found from the quadratic formula using the discriminant D = b² - 4ac. If D > 0, there are two real roots; if D = 0, there is one repeated root; if D < 0 the quadratic has no real roots and is irreducible over the reals.

Enter your quadratic

Coefficients (ax² + bx + c)

For example, for 2x² − 3x − 5, use a = 2, b = −3, c = −5.

Options:

Show step-by-step solution

Highlight when factoring over integers is possible

Show quadratic graph with x-intercepts

Quick picks:

Result:

No results yet. Enter a, b, c and click Factor.

How this calculator works

We first check that a ≠ 0 so the expression is truly quadratic.
We compute the discriminant D = b² − 4ac.
We find roots using the quadratic formula and, when possible, write the quadratic in factored form a(x − r₁)(x − r₂).
If the roots are integers or simple fractions, we highlight factorization over the integers or rationals. If D < 0, the quadratic cannot be factored over the real numbers.

Key formulas

Quadratic form: ax² + bx + c

Discriminant: D = b² − 4ac

Quadratic formula: x = [−b ± √(b² − 4ac)] / (2a)

Factored form (when possible): a(x − r₁)(x − r₂), where r₁ and r₂ are the roots.

Example problems and step-by-step solutions

Example 1 — Factor x² + 5x + 6

Here a = 1, b = 5, c = 6.
D = 5² − 4·1·6 = 25 − 24 = 1.
Roots: x = [−5 ± √1] / 2 = (−5 ± 1) / 2 → x = −2, x = −3.
So x² + 5x + 6 = (x + 2)(x + 3).

Example 2 — Factor 2x² − 3x − 5

a = 2, b = −3, c = −5.
D = (−3)² − 4·2·(−5) = 9 + 40 = 49.
Roots: x = [3 ± √49] / 4 = (3 ± 7) / 4 → x = 10/4 = 5/2, x = −4/4 = −1.
So 2x² − 3x − 5 = 2(x − 5/2)(x + 1).
You can also write it as (2x − 5)(x + 1).

Example 3 — x² + 2x + 5 (irreducible over ℝ)

a = 1, b = 2, c = 5.
D = 2² − 4·1·5 = 4 − 20 = −16 < 0.
The roots are complex, so x² + 2x + 5 does not factor over the real numbers; it is irreducible over ℝ.

Frequently asked questions

Q: What does the Factoring Calculator do?

It takes a quadratic of the form ax² + bx + c, computes the discriminant and roots, and then writes the expression in factored form when possible. It also shows each step so you can follow the algebra.

Q: When can a quadratic be factored over the real numbers?

A quadratic can be factored over the real numbers when its discriminant D = b² − 4ac is greater than or equal to zero. If D > 0, there are two distinct real roots; if D = 0, there is one repeated root; if D < 0, there are no real roots and the quadratic is irreducible over the reals.

Q: What if the quadratic does not factor nicely with integers?

The calculator still uses the quadratic formula to find the roots and can express them using fractions or square roots. It also tells you when a factorization would require complex numbers rather than real factors.

No practice sets found.