Factoring Calculator
Factor quadratic expressions of the form ax² + bx + c. Find the factored form, roots, and discriminant.
Enter coefficients for ax² + bx + c
Enter coefficients to factor the expression.
Factoring Formula
ax² + bx + c = a(x − r₁)(x − r₂)
where r₁, r₂ = (−b ± √(b² − 4ac)) / 2a
where r₁, r₂ = (−b ± √(b² − 4ac)) / 2a
Factoring a quadratic expression means writing it as a product of two binomials. The roots (r₁ and r₂) are the values of x that make the expression equal zero.
Factoring Methods
- Simple factoring: Find two numbers that multiply to ac and add to b.
- Quadratic formula: Use when simple factoring isn't obvious.
- Completing the square: Rewrite in vertex form first.
Example: Factor x² − 5x + 6
Find two numbers that multiply to 6 and add to −5: −2 and −3
Factored form: (x − 2)(x − 3)
Roots: x = 2 and x = 3
Find two numbers that multiply to 6 and add to −5: −2 and −3
Factored form: (x − 2)(x − 3)
Roots: x = 2 and x = 3
The Discriminant
Δ = b² − 4ac
- Δ > 0: Two distinct real roots (factorable over reals)
- Δ = 0: One repeated real root (perfect square)
- Δ < 0: No real roots (not factorable over reals)
FAQ
What if the expression can't be factored?
If the discriminant is negative, the quadratic has no real roots and cannot be factored using real numbers. It can still be factored using complex numbers.
How do I factor when a ≠ 1?
Use the AC method: multiply a × c, find two numbers that multiply to ac and add to b, then regroup and factor by grouping.
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Last updated: 2026-08-08