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Factoring

General

Factoring (Algebraic)

The quickest method for solving a quadratic equation when its roots are integers or simple fractions, rewriting the equation as a product of two linear factors and applying the zero-product property.

Definition

Factoring is a method for solving a quadratic equation by rewriting it as a product of two linear expressions, then applying the zero-product property, if two factors multiply to zero, at least one of them must equal zero. It's the fastest solving method when a quadratic's roots happen to be integers or simple fractions, though it doesn't work cleanly for every equation.

Unlike completing the square or the quadratic formula, which work for any quadratic regardless of how messy the numbers are, factoring by inspection depends on being able to spot the right integer or fraction pair, a skill that gets faster with practice but isn't guaranteed to work quickly for every equation.

Formula

ax² + bx + c = 0 → a(x − p)(x − q) = 0

Where p and q are the equation's roots, found by identifying two numbers that multiply to give a×c and add to give b.

Worked Example

Solve x² + 7x + 12 = 0 by factoring:

  1. Find two numbers that multiply to 12 and add to 7: those numbers are 3 and 4
  2. Rewrite: (x + 3)(x + 4) = 0
  3. Apply the zero-product property: x + 3 = 0 or x + 4 = 0
  4. Solve: x = −3 or x = −4

Checking both values against the original equation confirms they satisfy x² + 7x + 12 = 0.

Key Things to Know

  • The discriminant predicts whether clean factoring is even possible. A perfect square discriminant means rational factors exist, otherwise factoring by inspection will be frustrating or impossible.
  • Look for two numbers that multiply to a×c and add to b. This is the core pattern-matching step for factoring a quadratic where the leading coefficient a isn't 1.
  • Not every quadratic factors nicely, and that's expected. Many real-world quadratics don't have clean integer roots, in those cases, switching to the quadratic formula is faster than continuing to hunt for factors.
  • Factored form directly reveals the roots. Once written as (x − p)(x − q) = 0, the solutions p and q are immediately visible, no further calculation needed.
  • Practice speeds up pattern recognition significantly. Spotting factor pairs quickly is a skill that improves with repetition, making factoring genuinely faster than other methods once it's familiar.

Related Calculators

Quadratic Formula Calculator →Solve any quadratic equation instantly. Enter coefficients a, b, and c to find real and complex roots with step-by-step working shown for free.

Related Terms

Quadratic EquationQuadratic EquationCompleting the SquareCompleting the Square

Frequently Asked Questions

How do I know if a quadratic equation will factor nicely?
Check the [discriminant](/glossary/discriminant/), b² − 4ac, if it's a perfect square, the equation factors into clean rational numbers. If not, factoring by inspection will be difficult or impossible, and completing the square or the quadratic formula is the better route.
What's the zero-product property, and why does it matter for factoring?
It states that if two factors multiply to zero, at least one of them must be zero. Once a quadratic is rewritten as (x − p)(x − q) = 0, this property lets you set each factor to zero separately and solve directly.
Is factoring always faster than using the quadratic formula?
Only when the factors are easy to spot, for equations with large or non-integer coefficients, hunting for factors can take longer than simply applying the quadratic formula directly, which always works regardless of how messy the numbers are.
Can every quadratic equation be factored?
Not into simple integer or rational factors, many quadratics only factor using irrational or complex numbers, which isn't practically useful for solving by hand, that's when [completing the square](/glossary/completing-the-square/) or the quadratic formula become the better tools.
What's the relationship between factoring and the roots of a quadratic?
If a quadratic factors as (x − p)(x − q) = 0, then p and q are exactly the equation's solutions, the roots. This is the direct link between the factored form and the answer you're solving for.