Quadratic Functions

Solving by Factoring

Solving a quadratic equation by factoring and using the zero product property.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

When a Product Equals Zero

Before reading on, think about this: if two numbers multiply together to give exactly 0, what must be true about at least one of those two numbers? Is there any way for two genuinely nonzero numbers to multiply to 0?

Definition — Zero Product Property

If the product of two factors equals 0, then at least one of the factors must equal 0. For real numbers, there's no other way to reach a product of exactly 0.

This property is exactly what makes factoring useful for solving equations, not just simplifying expressions. Once a quadratic equation is written as (factor 1)(factor 2) = 0, the zero product property guarantees that setting each factor equal to 0 separately finds every solution — the equation's roots, also called its solutions or, when graphed, its x-intercepts (the points where the parabola crosses the x-axis, since y = 0 there).

Worked Example — Solving from Already-Factored Form

Solve (x − 4)(x + 2) = 0. By the zero product property, either x − 4 = 0 or x + 2 = 0. Solving each: x = 4 or x = −2. Check both: (4 − 4)(4 + 2) = 0 ✓, and (−2 − 4)(−2 + 2) = 0 ✓.

Worked Example — Factoring First, Then Solving

Solve x² − x − 12 = 0. Factor the trinomial: find two numbers multiplying to −12 and adding to −1: −4 and 3. Factored: (x − 4)(x + 3) = 0. Solve each factor: x = 4 or x = −3.

Worked Example — Rearranging Before Factoring

Solve x² + 5x = 14. The zero product property only applies when one side is 0, so rearrange first: x² + 5x − 14 = 0. Factor: two numbers multiplying to −14 and adding to 5 are 7 and −2, giving (x + 7)(x − 2) = 0. Solve: x = −7 or x = 2.

Tip

Always confirm one side of the equation is exactly 0 before factoring to solve — factoring an equation like x² − x = 12 without first moving the 12 over produces factors that don't actually correspond to the equation's real solutions.

Common Mistakes

Applying the zero product property to an equation before rearranging it so one side actually equals 0, such as trying to solve x(x + 3) = 10 by setting x = 10 and x + 3 = 10.
The zero product property only works when the product genuinely equals 0 — rearrange the entire equation to that form first, then factor, then apply the property.
Solving only one of the two factors and reporting a single solution, missing the second root.
A factored quadratic equation almost always has two factors, each producing its own solution — set every factor equal to 0 separately and solve each one.

Key Takeaways

The zero product property says a product can only equal 0 if at least one factor equals 0.
Solving a quadratic by factoring requires one side of the equation to be exactly 0 before factoring.
A quadratic's solutions are also called its roots or zeros, and correspond to the parabola's x-intercepts.

Summary

Factoring solves any quadratic equation that factors neatly, connecting roots directly to the parabola's x-intercepts. The final lesson provides a method that works even when a quadratic refuses to factor at all.

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