What this calculator does
This factoring calculator takes a quadratic trinomial in the form ax² + bx + c and rewrites it as a product of two linear factors, (x − r1)(x − r2), using the two roots of the equation. It is built specifically for factoring quadratics, not for breaking a plain number down into its prime factors, which is a different kind of problem covered by a separate prime factorisation tool.
Enter the three coefficients a, b and c and the calculator finds the discriminant, works out the roots, and reports back the factored form when one exists over the rational or real numbers. Not every quadratic factors neatly: the shape of the answer depends entirely on the sign and value of the discriminant.
The formula
The discriminant D = b² − 4ac decides what kind of answer is possible. If D is negative, the quadratic has no real roots and cannot be factored over the real numbers, only over complex numbers. If D is zero or a perfect square (with whole-number coefficients), the roots are rational and the calculator gives an exact factored form such as (x − 2)(x − 3). If D is positive but not a perfect square, the roots are real but irrational, so the calculator gives the exact roots and an approximate decimal factorisation instead of a clean rational one.
| Term | Meaning |
|---|---|
| a, b, c | The coefficients of the quadratic trinomial ax² + bx + c, with a the leading coefficient (never zero). |
| Discriminant (D) | D = b² − 4ac, the value that determines whether the roots are real, and whether they are rational. |
| Roots | The values of x where the quadratic equals zero: r = (−b ± √D) ÷ 2a. |
| Factored form | The quadratic rewritten as a(x − r1)(x − r2), which multiplies back out to the original expression. |
The inputs explained
| Field | What to enter |
|---|---|
| a | The leading coefficient, the number multiplying x². Must not be zero. |
| b | The coefficient multiplying x. |
| c | The constant term with no x attached. |
When to use it
Checking factoring homework
Entering a, b and c and comparing the calculator’s factored form against a hand-worked answer is a fast way to check a factoring exercise, including cases where the intended answer involves fractions.
Finding the roots of a quadratic equation
The roots shown alongside the factored form are exactly the x-intercepts of the parabola y = ax² + bx + c, useful whenever the actual goal is solving ax² + bx + c = 0 rather than the factored expression itself.
Working out whether a quadratic factors at all
Before spending time trying to factor a trinomial by hand, checking the discriminant here shows immediately whether a clean rational factoring exists, or whether the expression only factors with irrational or complex numbers.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How the factored form changes as the constant term changes, with rational roots throughout
The leading coefficient fixed at 1 and b fixed at 7, across a range of constant terms chosen so the discriminant stays a perfect square and the roots stay rational.
| Constant term (c) | Factored form | Roots | Discriminant |
|---|---|---|---|
| -8 | (x − 1)(x + 8) | x = 1 or x = -8 | 81.000: two rational roots |
| 0 | (x − 0)(x + 7) | x = 0 or x = -7 | 49.000: two rational roots |
| 6 | (x + 1)(x + 6) | x = -1 or x = -6 | 25.000: two rational roots |
| 10 | (x + 2)(x + 5) | x = -2 or x = -5 | 9.000: two rational roots |
| 12 | (x + 3)(x + 4) | x = -3 or x = -4 | 1.000: two rational roots |
What happens when the discriminant is negative
The leading coefficient fixed at 1 and b fixed at 2, across a range of constant terms chosen so the discriminant stays negative and no real factoring exists.
| Constant term (c) | Factored form | Discriminant | Roots |
|---|---|---|---|
| 2 | Does not factor over the real numbers (discriminant is negative) | -4.000: no real roots | Complex: -1.000 ± 1.000i |
| 3 | Does not factor over the real numbers (discriminant is negative) | -8.000: no real roots | Complex: -1.000 ± 1.414i |
| 5 | Does not factor over the real numbers (discriminant is negative) | -16.000: no real roots | Complex: -1.000 ± 2.000i |
| 8 | Does not factor over the real numbers (discriminant is negative) | -28.000: no real roots | Complex: -1.000 ± 2.646i |
| 12 | Does not factor over the real numbers (discriminant is negative) | -44.000: no real roots | Complex: -1.000 ± 3.317i |
Questions
Is this the same as a prime factorisation calculator?
No. This factoring calculator factors a quadratic trinomial, an algebraic expression with an x² term, into two linear factors. Breaking a plain whole number down into its prime factors is a different operation, covered by the separate prime factorisation calculator on this site.
What does it mean when the calculator says the quadratic does not factor?
A negative discriminant means the quadratic has no real roots at all, so it cannot be written as a product of real linear factors, only complex ones. A positive but non-perfect-square discriminant means real roots exist but they are irrational, so the exact factored form cannot be written with whole numbers or simple fractions.
Why does the factored form sometimes contain fractions?
When the leading coefficient a is not 1, the roots are often fractions rather than whole numbers, and the factoring calculator represents them exactly as fractions rather than rounding them, since a fraction gives the precise factored form while a decimal would only approximate it.
How does the discriminant relate to the shape of the parabola?
A positive discriminant means the parabola crosses the x-axis at two points, matching two real roots. A zero discriminant means it touches the axis at exactly one point, a repeated root. A negative discriminant means the parabola never crosses the x-axis at all, which is why no real factoring exists in that case.
To solve ax² + bx + c = 0 directly, including the same discriminant and root logic, see the quadratic equation calculator. For breaking a whole number down into its prime factors instead of factoring an expression, use the prime factorisation calculator.