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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. 415 (page 736)

In the following exercises, perform the indicated operations
$\frac{2 x}{x^{2} + 10 x + 24} + \frac{3 x}{x^{2} + 8 x + 16}$

Short Answer

Expert verified

Simplified expression is $\frac{5 x^{2} + 26 x}{{(x + 4)}^{2} (x + 6)}$

Step by step solution

01

Step 1. Given information

Expression:

\frac{2 x}{x^{2} + 10 x + 24} + \frac{3 x}{x^{2} + 8 x + 16}

02

Step 2. Find LCD

Denominator of first fraction is $x^{2} + 10 x + 24$

Factorize this denominator

$x^{2} + 6 x + 4 x + 24 = x (x + 6) + 4 (x + 6) = (x + 6) (x + 4)$

Denominator of the second fraction is $x^{2} + 8 x + 16$

Factorize this denominator

$x^{2} + 2.4 . x + 4^{2} = {(x + 4)}^{2}$

here $(x + 4)$ is common between both denominator

So the LCD is ${(x + 4)}^{2} (x + 6)$

03

Step 3. Rewrite each rational expression as an equivalent rational expression with the LCD

$\frac{2 x (x + 4)}{{(x + 4)}^{2} (x + 6)} + \frac{3 x (x + 6)}{{(x + 4)}^{2} (x + 6)}$

04

Step 4.Simplify the numerators

$\frac{2 x^{2} + 8 x}{{(x + 4)}^{2} (x + 6)} + \frac{3 x^{2} + 18 x}{{(x + 4)}^{2} (x + 6)}$

05

Step 5. Add numerators of the fractions

$\frac{2 x^{2} + 8 x + 3 x^{2} + 18 x}{{(x + 4)}^{2} (x + 6)}$

06

Step 6. Simplify numerator

$\frac{5 x^{2} + 26 x}{{(x + 4)}^{2} (x + 6)}$

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Most popular questions from this chapter

In the following exercises, solve each rational inequality and write the solution in interval notation.

$\frac{4}{x^{2} - 25} > 0$

For the following exercises, perform the indicated operations.
$\frac{12 p^{2} + 3 p}{p + 3} 梅 \frac{p^{2} + 2 p - 63}{p^{2} - p - 12} . \frac{p - 7}{9 p^{3} - 9 p^{2}}$

Simplify the complex rational expression by using the LCD:

$\frac{\frac{1}{a} + \frac{1}{b}}{\frac{a}{b} + \frac{b}{a}}$

In the following exercise, find

(a) $R (x) = f (x) + g (x)$

(b) $R (x) = f (x) - g (x)$ :

$f (x) = \frac{6 x}{x^{2} - 64} g (x) = \frac{3}{x - 8}$

In the following exercises, divide the rational expressions.

$\frac{\frac{3 b^{2} + 2 b - 8}{12 b + 18}}{\frac{3 b^{2} + 2 b - 8}{2 b^{2} - 7 b - 15}}$

See all solutions

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