1Cademy - Solving $$(5n - 2)(6n - 1) = 0$$ Using the Zero Product Property

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Quadratic Equation

Example

Solving $(5n - 2)(6n - 1) = 0$ Using the Zero Product Property

Solve $(5n - 2)(6n - 1) = 0$ by applying the Zero Product Property. This example involves multi-step linear equations whose solutions are fractions.

Step 1 — Set each factor equal to zero using the Zero Product Property:

$5n - 2 = 0 \quad \text{or} \quad 6n - 1 = 0$

Step 2 — Solve each linear equation. Unlike the simplest cases where a single operation isolates the variable, each equation here requires two steps (adding a constant and then dividing by the coefficient):

$5n = 2 \implies n = \frac{2}{5} \quad \text{or} \quad 6n = 1 \implies n = \frac{1}{6}$

Step 3 — Check both solutions by substituting each value into the original equation:

For $n = \frac{2}{5}$ : $\left(5 \cdot \frac{2}{5} - 2\right)\left(6 \cdot \frac{2}{5} - 1\right) = (0)\left(\frac{7}{5}\right) = 0$ ✓

For $n = \frac{1}{6}$ : $\left(5 \cdot \frac{1}{6} - 2\right)\left(6 \cdot \frac{1}{6} - 1\right) = \left(-\frac{7}{6}\right)(0) = 0$ ✓

The solutions are $n = \frac{2}{5}$ and $n = \frac{1}{6}$ . Each solution makes exactly one factor equal to zero while the other factor is nonzero, yet the product is zero in both cases.

Updated 2026-04-21

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OpenStax Elementary Algebra

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