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Chapter 7: Problem 52

Add or subtract as indicated. Simplify the result, if possible. $$\frac{6}{x^{2}-4}+\frac{2}{(x+2)^{2}}$$

Short Answer

Expert verified
The result is \(\frac{8x+8}{(x-2)(x+2)^2}\)

Step by step solution

01

Factor the Denominator of the First Fraction

The denominator of the first fraction can be factored as \(x^{2}-4=(x-2)(x+2)\). Therefore, the first fraction can be rewritten as \(\frac{6}{(x-2)(x+2)}\)
02

Find a Common Denominator

Using the factored form of the first fraction, the common denominator would be \((x-2)(x+2)^2\). Therefore, the fractions can be rewritten with this common denominator: \(\frac{6(x+2)}{(x-2)(x+2)^2}+\frac{2(x-2)}{(x-2)(x+2)^2}\)
03

Add the Fractions

With the same denominators, we can add the fractions: \(\frac{6(x+2)+2(x-2)}{(x-2)(x+2)^2}\)
04

Simplify the Result

Simplify the result to get: \(\frac{6x+12+2x-4}{(x-2)(x+2)^2} = \frac{8x+8}{(x-2)(x+2)^2}\)

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Most popular questions from this chapter

perform the indicated operation or operations. Simplify the result, if possible. $$\frac{6 b^{2}-10 b}{16 b^{2}-48 b+27}+\frac{7 b^{2}-20 b}{16 b^{2}-48 b+27}-\frac{6 b-3 b^{2}}{16 b^{2}-48 b+27}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.. $$Find the missing polynomials: $\quad-\frac{3 x-12}{2 x}=\frac{3}{2}$$

Explain how to add rational expressions when denominators are the same. Give an example with your explanation.

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{x^{2}}{x-3}+\frac{9}{3-x}$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I can solve \(\frac{x}{9}=\frac{4}{6}\) by using the cross-products principle or by multiplying both sides by \(18,\) the least common denominator.
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