1Cademy - Subtracting $$\frac{y^2}{y-6}

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How to Add and Subtract Rational Expressions with a Common Denominator

Example

Subtracting $\frac{y^2}{y-6} - \frac{2y+24}{y-6}$

Subtract two rational expressions that share the denominator $y - 6$ , where the second numerator is a binomial that requires careful sign distribution:

$\frac{y^2}{y - 6} - \frac{2y + 24}{y - 6}$

Step 1 — Subtract the numerators over the common denominator. Since both fractions share $y - 6$ as their denominator, subtract the numerators. Because the second numerator is a binomial, place it in parentheses to ensure the subtraction applies to every term:

$\frac{y^2 - (2y + 24)}{y - 6}$

Step 2 — Distribute the negative sign in the numerator. Multiply each term inside the parentheses by $-1$ , changing $+2y$ to $-2y$ and $+24$ to $-24$ :

$\frac{y^2 - 2y - 24}{y - 6}$

Step 3 — Factor the numerator. The trinomial $y^2 - 2y - 24$ factors as $(y - 6)(y + 4)$ , since $(-6) + 4 = -2$ and $(-6) \cdot 4 = -24$ :

$\frac{(y - 6)(y + 4)}{y - 6}$

Step 4 — Simplify by dividing out the common factor. Cancel the shared factor $(y - 6)$ from the numerator and denominator:

$y + 4$

This example highlights a critical step that does not arise in addition problems: when the numerator being subtracted contains more than one term, parentheses must be used and the negative sign must be distributed to every term in that numerator. Forgetting to distribute the negative sign — for instance, writing $y^2 - 2y + 24$ instead of $y^2 - 2y - 24$ — is a common error that produces an incorrect result.

Updated 2026-04-21

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OpenStax Elementary Algebra

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