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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. 7.36 (page 661)

Add:
$\frac{2 n}{n^{2} - 3 n - 10} + \frac{6}{n^{2} + 5 n + 6}$

Short Answer

Expert verified

The required sum is $= \frac{2 n^{2} + 6 n - 24}{(n - 5) (n + 2) (n + 3)}$ .

Step by step solution

01

Step 1. Given information

The given expression is $\frac{2 n}{n^{2} - 3 n - 10} + \frac{6}{n^{2} + 5 n + 6}$

02

Step 2. Find LCD

Factor the denominators of both the rational expressions.

$\begin{array}{rcl} n^{2} - 3 n - 10 & = & (n - 5) (n + 2) \\ n^{2} + 5 n + 6 & = & (n + 3) (n + 2) \end{array}$

So, the LCD is $(n - 5) (n + 2) (n + 3)$ .

03

Step 3. Express the expression with LCD

Rewrite each rational expression in terms of LCD.

$\begin{array}{rcl} \frac{2 n}{n^{2} - 3 n - 10} & = & \frac{2 n^{2} + 6}{(n - 5) (n + 2) (n + 3)} \\ \frac{6}{n^{2} + 5 n + 6} & = & \frac{6 n - 30}{(n - 5) (n + 2) (n + 3)} \end{array}$

04

Step 4 / 4Step 4. Add the obtained Expressions

Add the expressions with LCD.

$\begin{array}{rcl} \frac{2 n^{2} + 6}{(n - 5) (n + 2) (n + 3)} + \frac{6 n - 30}{(n - 5) (n + 2) (n + 3)} & = & \frac{2 n^{2} + 6 + 6 n - 30}{(n - 5) (n + 2) (n + 3)} \\ = & \frac{2 n^{2} + 6 n - 24}{(n - 5) (n + 2) (n + 3)} \end{array}$

So, the required LCD is $\frac{2 n^{2} + 6 n - 24}{(n - 5) (n + 2) (n + 3)}$

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