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Chapter 0: Problem 45

Add or subtract as indicated. $$\frac{2 x}{x+2}+\frac{x+2}{x-2}$$

Short Answer

Expert verified
The solution is \(\frac{3 x^2+4}{(x+2)(x-2)}\)

Step by step solution

01

Find the least common denominator (LCD)

Determine the least common denominator. Since the denominators \(x+2\) and \(x-2\) have no common factors other than 1, the least common denominator is simply the product of the two denominators, which is \((x+2)(x-2)\).
02

Make the denominators of both fractions equal to the LCD

Rewrite each fraction so that the denominator is equal to the LCD. For the first term \(\frac{2 x}{x+2}\) to have the LCD, it needs a factor of \(x-2\) in its denominator. This can be done by multiplying both top and bottom by \(x-2\), which gives: \[\frac{2 x*(x-2)}{(x+2)(x-2)}\] In the same way, the second term \(\frac{x+2}{x-2}\) is rewritten to have the LCD by multiplying top and bottom by \(x+2\), which gives: \[\frac{(x+2)^2}{(x+2)(x-2)}\]
03

Perform the operations in the numerator

Perform the multiplications in the numerators of the two terms. This results in: \[\frac{2 x^2-4 x}{(x+2)(x-2)}+\frac{x^2+4x+4}{(x+2)(x-2)}\]
04

Add the fractions

Since the denominators are the same, the fractions can be added together by adding their numerators: \[\frac{2 x^2-4 x+x^2+4x+4}{(x+2)(x-2)}\]
05

Simplify the result

Simplify the above fraction by combining like terms in the numerator, which yields: \[\frac{3 x^2+4}{(x+2)(x-2)}\]

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Most popular questions from this chapter

In Exercises \(133-136\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$7^{\frac{1}{2}} \cdot 7^{\frac{1}{2}}=49$$

In Exercises \(133-136\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{\sqrt{20}}{8}=\frac{\sqrt{10}}{4}$$

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Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. To earn an A in a course, you must have a final average of at least \(90 \% .\) On the first four examinations, you have grades of \(86 \%, 88 \%, 92 \%,\) and \(84 \% .\) If the final examination counts as two grades, what must you get on the final to earn an A in the course?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{x^{2}-25}{x-5}=x-5$$
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