Problem 52 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \(\frac{6}{(x-2)(x+2)}\)

Step 2: 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: \(\frac{6(x+2)}{(x-2)(x+2)^2}+\frac{2(x-2)}{(x-2)(x+2)^2}\)

Step 3: Add the Fractions

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

Step 4: Simplify the Result

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