Simplify 9/(x+8)-(8x)/(x^2-64)

$\frac{9}{x + 8} - \frac{8 x}{x^{2} - 64}$

Step 1

Simplify the denominator.

Rewrite $64$ as $8^{2}$ .

$\frac{9}{x + 8} - \frac{8 x}{x^{2} - 8^{2}}$

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 $b = 8$ .

$\frac{9}{x + 8} - \frac{8 x}{(x + 8) (x - 8)}$

Step 2

To write $\frac{9}{x + 8}$ as a fraction with a common denominator, multiply by $\frac{x - 8}{x - 8}$ .

$\frac{9}{x + 8} \cdot \frac{x - 8}{x - 8} - \frac{8 x}{(x + 8) (x - 8)}$

Step 3

Simplify terms.

Multiply $\frac{9}{x + 8}$ by $\frac{x - 8}{x - 8}$ .

$\frac{9 (x - 8)}{(x + 8) (x - 8)} - \frac{8 x}{(x + 8) (x - 8)}$

Combine the numerators over the common denominator.

$\frac{9 (x - 8) - 8 x}{(x + 8) (x - 8)}$

Step 4

Simplify the numerator.

Apply the distributive property.

$\frac{9 x + 9 \cdot - 8 - 8 x}{(x + 8) (x - 8)}$

Multiply $9$ by $- 8$ .

$\frac{9 x - 72 - 8 x}{(x + 8) (x - 8)}$

Subtract $8 x$ from $9 x$ .

$\frac{x - 72}{(x + 8) (x - 8)}$