Factoring Quadratics Calculator
Enter the coefficients a, b, and c to factor a quadratic into its root-based form a(x − r₁)(x − r₂).
Non-negative: real factorization exists
- 1
b² (b squared)
-5 × -5 = 25 - 2
4ac
4 × 1 × 6 = 24 - 3
Discriminant D = b² − 4ac
25 − 24 = 1
How does this calculator work?
A quadratic ax² + bx + c factors as a(x − r₁)(x − r₂), where the roots r₁ and r₂ come from r = (−b ± √(b² − 4ac)) / (2a). If the discriminant b² − 4ac is negative, the quadratic is irreducible over the reals and only complex roots exist.
Formula
How this is calculated
A quadratic ax² + bx + c factors as a(x − r₁)(x − r₂), where r₁ and r₂ are its roots. The leading coefficient a (which must be nonzero) scales the parabola, while b and c shift it. This calculator finds the roots with the quadratic formula r = (−b ± √D) / (2a), where D = b² − 4ac is the discriminant.
The discriminant determines the nature of the factorization. When D > 0 there are two distinct real roots and the trinomial factors into two distinct linear terms. When D = 0 there is one repeated root, giving a perfect-square factor a(x − r)². When D < 0 the roots are complex conjugates, so the quadratic is irreducible over the real numbers; the calculator then reports the complex roots r = (−b ± √(−D) i) / (2a).
Coefficients are unitless. Results are rounded for display, so a factored form may differ slightly from an exact integer factorization when the roots are irrational. Setting a = 0 is rejected because the expression is then linear, not quadratic.
Frequently asked questions
A negative discriminant means the quadratic has no real roots and cannot be factored over the real numbers. The calculator instead reports the pair of complex-conjugate roots.
If a = 0 the expression bx + c is linear, not quadratic, so there is no a(x − r₁)(x − r₂) factorization to compute.
When the roots are irrational or non-integer, the factors involve those exact root values, which are shown as rounded decimals rather than tidy integers.
Also known as
TG we-Calculate Editorial Team. (2026). Factoring Quadratics Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/factoring-quadratics-calculator
TG we-Calculate Editorial Team. "Factoring Quadratics Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/factoring-quadratics-calculator.
TG we-Calculate Editorial Team, "Factoring Quadratics Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/factoring-quadratics-calculator
@misc{wecalculate_factoring_quadratics_calculator, title = {Factoring Quadratics Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/factoring-quadratics-calculator}}, year = {2026}, note = {TG we-Calculate} }
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