Overview
Factoring a quadratic means writing it as a product of two linear factors. Factoring is useful for solving equations quickly, finding roots and understanding the parabola. Not every quadratic can be factored over the real numbers.
Formula
ax² + bx + c = a(x − r₁)(x − r₂), where r₁ and r₂ are the roots.
How it works
Compute the discriminant
Δ = b² − 4ac determines the root type.
Find two numbers
Find two numbers whose product is ac and whose sum is b.
Write the factors
Arrange them as (px + q)(rx + s).
Verify
Multiply the factors and confirm you recover the original polynomial.
Worked examples
Example 1: x² − 5x + 6
The factors are (x − 2)(x − 3), since −2 − 3 = −5 and (−2)(−3) = 6.
Example 2: x² + 4x + 4
A perfect square: (x + 2)².
Frequently asked questions
How do I know the factorization is correct?
Multiply the factors; you should recover the original polynomial.
What if there are no integer factors?
Use the real roots or the quadratic formula, or note that no real factorization exists.
What does perfect square mean?
The two factors are identical, such as (x + 2)(x + 2) = (x + 2)².
This calculator is for educational and informational purposes only and is not a substitute for professional advice.