Problem 45 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Find Common Denominator and Rewrite the Fractions

The common denominator here is \((x+2)(x-2)\). So, rewrite the given expression as \(\frac{2 x(x - 2)}{(x + 2)(x - 2)} + \frac{(x + 2)(x+2)}{(x + 2)(x - 2)}\).

Step 2: Simplify the Numerators

Rewriting each fraction with the common denominator yields \(\frac{2x^2 - 4x}{(x + 2)(x - 2)} + \frac{x^2 + 4x + 4}{(x + 2)(x - 2)}\). This can be simplified into a single fraction: \(\frac{2x^2 - 4x + x^2 + 4x + 4}{(x + 2)(x - 2)}\).

Step 3: Combine Like Terms and Simplify

Combining like terms in the numerator results in: \(\frac{3x^2 + 4}{(x + 2)(x - 2)}\) which is the final form of the expression.