Simplify 2/(x-4)+4/(x^2-16)

$\frac{2}{x - 4} + \frac{4}{x^{2} - 16}$

Step 1

Simplify the denominator.

Rewrite $16$ as $4^{2}$ .

$\frac{2}{x - 4} + \frac{4}{x^{2} - 4^{2}}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 4$ .

$\frac{2}{x - 4} + \frac{4}{(x + 4) (x - 4)}$

Step 2

To write $\frac{2}{x - 4}$ as a fraction with a common denominator, multiply by $\frac{x + 4}{x + 4}$ .

$\frac{2}{x - 4} \cdot \frac{x + 4}{x + 4} + \frac{4}{(x + 4) (x - 4)}$

Step 3

Write each expression with a common denominator of $(x - 4) (x + 4)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{2}{x - 4}$ by $\frac{x + 4}{x + 4}$ .

$\frac{2 (x + 4)}{(x - 4) (x + 4)} + \frac{4}{(x + 4) (x - 4)}$

Reorder the factors of $(x - 4) (x + 4)$ .

$\frac{2 (x + 4)}{(x + 4) (x - 4)} + \frac{4}{(x + 4) (x - 4)}$

Step 4

Combine the numerators over the common denominator.

$\frac{2 (x + 4) + 4}{(x + 4) (x - 4)}$

Step 5

Simplify the numerator.

Factor $2$ out of $2 (x + 4) + 4$ .

Factor $2$ out of $4$ .

$\frac{2 (x + 4) + 2 \cdot 2}{(x + 4) (x - 4)}$

Factor $2$ out of $2 (x + 4) + 2 \cdot 2$ .

$\frac{2 (x + 4 + 2)}{(x + 4) (x - 4)}$

Add $4$ and $2$ .

$\frac{2 (x + 6)}{(x + 4) (x - 4)}$