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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 7: Q. 7.28 (page 657)

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

Short Answer

Expert verified

The required result is $\frac{(x - 3)}{(x + 9)}$ .

Step by step solution

01

Step 1. Given Information

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

02

Step 2. Subtract the Numerators

As the denominator is same for both the terms, add the numerator and place over the common denominator.
Add the like terms of the numerator.

$\begin{array}{rcl} \frac{6 x^{2} - x + 20}{x^{2} - 81} - \frac{5 x^{2} + 11 x - 7}{x^{2} - 81} & = & \frac{6 x^{2} - x + 20 - (5 x^{2} + 11 x - 7)}{x^{2} - 81} \\ = & \frac{6 x^{2} - x + 20 - 5 x^{2} - 11 x + 7}{x^{2} - 81} \\ = & \frac{x^{2} - 12 x + 27}{x^{2} - 81} \end{array}$

03

Step 3. Factor the Numerator

Factor the expression in the numerator and then cancel out the common terms from numerator and denominator.

$\begin{array}{rcl} \frac{x^{2} - 12 x + 27}{x^{2} - 81} & = & \frac{(x - 9) (x - 3)}{(x - 9) (x + 9)} \\ = & \frac{(x - 9) (x - 3)}{(x - 9) (x + 9)} \\ = & \frac{(x - 3)}{(x + 9)} \end{array}$

So, the resultant expression is $\frac{(x - 3)}{(x + 9)}$ .

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