Simplify ((x^2-36)/(8x+48))÷((x^2-4x-12)/(2x))

$\frac{x^{2} - 36}{8 x + 48} \div \frac{x^{2} - 4 x - 12}{2 x}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 36}{8 x + 48} \cdot \frac{2 x}{x^{2} - 4 x - 12}$

Step 2

Simplify the numerator.

Rewrite $36$ as $6^{2}$ .

$\frac{x^{2} - 6^{2}}{8 x + 48} \cdot \frac{2 x}{x^{2} - 4 x - 12}$

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 $b = 6$ .

$\frac{(x + 6) (x - 6)}{8 x + 48} \cdot \frac{2 x}{x^{2} - 4 x - 12}$

Step 3

Simplify terms.

Factor $8$ out of $8 x + 48$ .

Factor $8$ out of $8 x$ .

$\frac{(x + 6) (x - 6)}{8 (x) + 48} \cdot \frac{2 x}{x^{2} - 4 x - 12}$

Factor $8$ out of $48$ .

$\frac{(x + 6) (x - 6)}{8 x + 8 \cdot 6} \cdot \frac{2 x}{x^{2} - 4 x - 12}$

Factor $8$ out of $8 x + 8 \cdot 6$ .

$\frac{(x + 6) (x - 6)}{8 (x + 6)} \cdot \frac{2 x}{x^{2} - 4 x - 12}$

Cancel the common factor of $2$ .

Factor $2$ out of $8 (x + 6)$ .

$\frac{(x + 6) (x - 6)}{2 (4 (x + 6))} \cdot \frac{2 x}{x^{2} - 4 x - 12}$

Factor $2$ out of $2 x$ .

$\frac{(x + 6) (x - 6)}{2 (4 (x + 6))} \cdot \frac{2 (x)}{x^{2} - 4 x - 12}$

Cancel the common factor.

$\frac{(x + 6) (x - 6)}{2 (4 (x + 6))} \cdot \frac{2 x}{x^{2} - 4 x - 12}$

Rewrite the expression.

$\frac{(x + 6) (x - 6)}{4 (x + 6)} \cdot \frac{x}{x^{2} - 4 x - 12}$

Multiply $\frac{(x + 6) (x - 6)}{4 (x + 6)}$ by $\frac{x}{x^{2} - 4 x - 12}$ .

$\frac{(x + 6) (x - 6) x}{4 (x + 6) (x^{2} - 4 x - 12)}$

Cancel the common factor of $x + 6$ .

Cancel the common factor.

$\frac{(x + 6) (x - 6) x}{4 (x + 6) (x^{2} - 4 x - 12)}$

Rewrite the expression.

$\frac{(x - 6) x}{4 (x^{2} - 4 x - 12)}$

Step 4

Factor $x^{2} - 4 x - 12$ 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 $- 12$ and whose sum is $- 4$ .

$- 6, 2$

Write the factored form using these integers.

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

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

Step 5

Cancel the common factor of $x - 6$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify ((x^2-36)/(8x+48))÷((x^2-4x-12)/(2x))

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