59 Solving Quadratic Equations by Factoring

Solving Quadratic Equations by Factoring is the study of finding the solutions to a quadratic equation by rewriting it as a product of factors set equal to zero, then applying the zero product property to split the single quadratic equation into simpler linear equations solvable independently.

The Scope of Factoring as a Solving Method

Solving a quadratic equation by factoring is possible whenever the quadratic expression involved can be factored into binomials with rational (typically integer) coefficients, using the trinomial factoring and special pattern techniques already established. This method produces exact solutions directly, without approximation, and is generally the fastest solving technique available whenever the quadratic in question does, in fact, factor cleanly.

Preparing a Quadratic Equation for Factoring

Before factoring can be applied, the quadratic equation must be arranged so that one side equals zero, moving every term to one side using the addition and subtraction properties of equality, and the resulting expression should be simplified and, ideally, written in standard form with a positive leading coefficient.

This zero-on-one-side arrangement is not optional; the zero product property, on which this entire solving method depends, only applies when a product of factors is set equal to zero, not to any other value.

Factoring the Quadratic Expression

With the equation arranged in the form (quadratic expression) = 0, the quadratic expression is factored using whichever technique applies — extracting a greatest common factor, applying the monic or nonmonic trinomial methods, or recognizing a special pattern such as a difference of squares or a perfect square trinomial.

Solving with the Zero Product Property

Once the quadratic is written as a product of factors equal to zero, the zero product property states that this can only be true if at least one of the individual factors equals zero. This allows the single quadratic equation to be split into two separate linear equations, one for each factor, each solved independently using ordinary one-step or multi-step linear equation techniques.

Solving each linear equation gives x = -2 or x = -3, and both values are genuine solutions to the original quadratic equation, since a quadratic equation of degree 2 generally has up to two solutions.

Special Cases in Factored Quadratic Solutions

When a factored quadratic contains a repeated factor, such as (x - 5)² = 0, the zero product property still applies, but both branches of the split produce the identical equation x - 5 = 0, yielding a single repeated solution, x = 5, sometimes described as a solution of multiplicity two. When one of the factors in a factored quadratic is a monomial rather than a binomial, such as x(x + 4) = 0, the zero product property produces the equations x = 0 and x + 4 = 0 directly, with the first branch requiring no further solving step at all.

Verifying Quadratic Solutions Found by Factoring

Each solution obtained by factoring is verified by substituting it back into the original, unfactored quadratic equation and confirming that both sides evaluate to the same number, exactly as verifying any equation's solution; because a quadratic equation typically has two solutions, both must be checked independently, since one solution being correct does not guarantee the other is as well.

Diagnosing Errors in Solving Quadratics by Factoring

Common errors in this area include attempting to apply the zero product property to an equation that is not actually set equal to zero on one side, factoring the quadratic expression incorrectly and thereby producing incorrect candidate solutions, applying the zero product property to a sum or a difference of factors rather than a genuine product, and reporting only one of the two solutions produced by a properly factored quadratic, particularly when one of the two resulting linear equations is solved more quickly than the other and its counterpart is overlooked.