Simplify ((8x-56)/(x^2-49))÷((x-6)/(x^2+11x+28))

$\frac{8 x - 56}{x^{2} - 49} \div \frac{x - 6}{x^{2} + 11 x + 28}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{8 x - 56}{x^{2} - 49} \cdot \frac{x^{2} + 11 x + 28}{x - 6}$

Step 2

Factor $8$ out of $8 x - 56$ .

Factor $8$ out of $8 x$ .

$\frac{8 (x) - 56}{x^{2} - 49} \cdot \frac{x^{2} + 11 x + 28}{x - 6}$

Factor $8$ out of $- 56$ .

$\frac{8 x + 8 \cdot - 7}{x^{2} - 49} \cdot \frac{x^{2} + 11 x + 28}{x - 6}$

Factor $8$ out of $8 x + 8 \cdot - 7$ .

$\frac{8 (x - 7)}{x^{2} - 49} \cdot \frac{x^{2} + 11 x + 28}{x - 6}$

Step 3

Simplify the denominator.

Rewrite $49$ as $7^{2}$ .

$\frac{8 (x - 7)}{x^{2} - 7^{2}} \cdot \frac{x^{2} + 11 x + 28}{x - 6}$

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 $b = 7$ .

$\frac{8 (x - 7)}{(x + 7) (x - 7)} \cdot \frac{x^{2} + 11 x + 28}{x - 6}$

Step 4

Cancel the common factor of $x - 7$ .

Cancel the common factor.

$\frac{8 (x - 7)}{(x + 7) (x - 7)} \cdot \frac{x^{2} + 11 x + 28}{x - 6}$

Rewrite the expression.

$\frac{8}{x + 7} \cdot \frac{x^{2} + 11 x + 28}{x - 6}$

Step 5

Factor $x^{2} + 11 x + 28$ 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 $28$ and whose sum is $11$ .

$4, 7$

Write the factored form using these integers.

$\frac{8}{x + 7} \cdot \frac{(x + 4) (x + 7)}{x - 6}$

Step 6

Simplify terms.

Cancel the common factor of $x + 7$ .

Factor $x + 7$ out of $(x + 4) (x + 7)$ .

$\frac{8}{x + 7} \cdot \frac{(x + 7) (x + 4)}{x - 6}$

Cancel the common factor.

$\frac{8}{x + 7} \cdot \frac{(x + 7) (x + 4)}{x - 6}$

Rewrite the expression.

$8 \frac{x + 4}{x - 6}$

Combine $8$ and $\frac{x + 4}{x - 6}$ .

$\frac{8 (x + 4)}{x - 6}$

Simplify ((8x-56)/(x^2-49))÷((x-6)/(x^2+11x+28))

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