Problem 45 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The simplified form of \(\frac{2x}{x+2} + \frac{x+2}{x-2}\) is \(\frac{3x^2+4}{x^2-4}\).

Step 1: Find the Common Denominator

The common denominator of \(\frac{2x}{x+2}\) and \(\frac{x+2}{x-2}\) is the product of \(x+2\) and \(x-2\), which can be simplified as \(x^2 - 4\).

Step 2: Create Equivalent Fractions

Now, rewrite both fractions with the common denominator. This is done by multiplying the numerator and denominator of each fraction by the denominator of the other fraction:\[\frac{2x}{x+2} * \frac{x-2}{x-2} + \frac{x+2}{x-2} * \frac{x+2}{x+2}\]Simplify to get:\[\frac{2x(x-2)}{x^2-4} + \frac{(x+2)^2}{x^2-4}\]

Step 3: Combine the Fractions

Now that the fractions have the same denominator, they can be combined:\[\frac{2x(x-2)+(x+2)^2}{x^2-4}\]

Step 4: Simplify the Numerator

To further simplify, expand and collect like terms in the numerator:\[\frac{2x^2-4x+x^2+4x+4}{x^2-4}\]This simplifies to:\[\frac{3x^2+4}{x^2-4}\]