1Cademy - Subtracting $$\frac{6x^2-x+20}{x^2-81}

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How to Add and Subtract Rational Expressions with a Common Denominator

Example

Subtracting $\frac{6x^2-x+20}{x^2-81} - \frac{5x^2+11x-7}{x^2-81}$

Subtract two rational expressions that share the common denominator $x^2-81$ :

$\frac{6x^2-x+20}{x^2-81} - \frac{5x^2+11x-7}{x^2-81}$

Step 1 — Subtract the numerators over the common denominator. Place the second numerator in parentheses and write the difference over the shared denominator:

$\frac{6x^2-x+20 - (5x^2+11x-7)}{x^2-81}$

Step 2 — Distribute the negative sign in the numerator. Multiply each term inside the parentheses by $-1$ , changing $+5x^2$ to $-5x^2$ , $+11x$ to $-11x$ , and $-7$ to $+7$ :

$\frac{6x^2-x+20 - 5x^2 - 11x + 7}{x^2-81}$

Step 3 — Combine like terms in the numerator. Group the $x^2$ -terms: $6x^2 - 5x^2 = x^2$ . Group the $x$ -terms: $-x - 11x = -12x$ . Group the constants: $20 + 7 = 27$ :

$\frac{x^2-12x+27}{x^2-81}$

Step 4 — Factor both the numerator and the denominator. The numerator $x^2-12x+27$ factors as $(x-3)(x-9)$ , since $(-3) + (-9) = -12$ and $(-3)(-9) = 27$ . The denominator $x^2-81$ is a difference of squares and factors as $(x-9)(x+9)$ , since $x^2 = x^2$ and $81 = 9^2$ :

$\frac{(x-3)(x-9)}{(x-9)(x+9)}$

Step 5 — Simplify by removing the common factor. Cancel the shared factor $(x-9)$ from the numerator and denominator:

$\frac{x-3}{x+9}$

This example provides further practice with the procedure for subtracting rational expressions with a common denominator, highlighting the distribution of the negative sign and the use of the difference of squares pattern when factoring the denominator to simplify the final result.

Updated 2026-05-26

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OpenStax Intermediate Algebra

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