1Cademy - Solving $$5x^2 - 13x = 7x$$ by Factoring

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Quadratic Equation

Example

Solving $5x^2 - 13x = 7x$ by Factoring

Solve $5x^2 - 13x = 7x$ by rearranging into standard form and factoring out the GCF.

Step 1 — Write in standard form. Subtract $7x$ from both sides to collect all terms on the left:

$5x^2 - 13x - 7x = 0$

$5x^2 - 20x = 0$

Step 2 — Factor the left side. Both terms share a greatest common factor of $5x$ . Factor it out:

$5x(x - 4) = 0$

Step 3 — Apply the Zero Product Property. Set each factor equal to zero:

$5x = 0 \quad \text{or} \quad x - 4 = 0$

Step 4 — Solve each equation:

$x = 0 \quad \text{or} \quad x = 4$

Step 5 — Check both solutions by substituting into the original equation $5x^2 - 13x = 7x$ :

For $x = 0$ : $5(0)^2 - 13(0) = 0$ and $7(0) = 0$ , so $0 = 0$ ✓

For $x = 4$ : $5(4)^2 - 13(4) = 80 - 52 = 28$ and $7(4) = 28$ , so $28 = 28$ ✓

The solutions are $x = 0$ and $x = 4$ . This example illustrates a case where, after writing the equation in standard form, the quadratic expression has no constant term — only $5x^2 - 20x$ . When the constant term is absent, the factoring step involves extracting a GCF (here $5x$ ) rather than factoring a trinomial. One of the resulting solutions is $x = 0$ , which arises from the monomial factor.

Updated 2026-04-21

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OpenStax Elementary Algebra

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