Simplify x/(x+4)-(x+36)/(x^2-16)

$\frac{x}{x + 4} - \frac{x + 36}{x^{2} - 16}$

Simplify the denominator.

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

$\frac{x}{x + 4} - \frac{x + 36}{x^{2} - 4^{2}}$

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 $b = 4$ .

$\frac{x}{x + 4} - \frac{x + 36}{(x + 4) (x - 4)}$

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

$\frac{x}{x + 4} \cdot \frac{x - 4}{x - 4} - \frac{x + 36}{(x + 4) (x - 4)}$

Simplify terms.

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

$\frac{x (x - 4)}{(x + 4) (x - 4)} - \frac{x + 36}{(x + 4) (x - 4)}$

Combine the numerators over the common denominator.

$\frac{x (x - 4) - (x + 36)}{(x + 4) (x - 4)}$

Simplify the numerator.

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 4 - (x + 36)}{(x + 4) (x - 4)}$

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot - 4 - (x + 36)}{(x + 4) (x - 4)}$

Move $- 4$ to the left of $x$ .

$\frac{x^{2} - 4 \cdot x - (x + 36)}{(x + 4) (x - 4)}$

Apply the distributive property.

$\frac{x^{2} - 4 x - x - 1 \cdot 36}{(x + 4) (x - 4)}$

Multiply $- 1$ by $36$ .

$\frac{x^{2} - 4 x - x - 36}{(x + 4) (x - 4)}$

Subtract $x$ from $- 4 x$ .

$\frac{x^{2} - 5 x - 36}{(x + 4) (x - 4)}$

Factor $x^{2} - 5 x - 36$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 36$ and whose sum is $- 5$ .

$- 9, 4$

Write the factored form using these integers.

$\frac{(x - 9) (x + 4)}{(x + 4) (x - 4)}$

Cancel the common factor of $x + 4$ .

Cancel the common factor.

$\frac{(x - 9) (x + 4)}{(x + 4) (x - 4)}$

Rewrite the expression.

$\frac{x - 9}{x - 4}$

Simplify x/(x+4)-(x+36)/(x^2-16)

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