Simplify ((-4x-12)/(x-7))÷((x^2-12x+36)/(2x-14))

$\frac{- 4 x - 12}{x - 7} \div \frac{x^{2} - 12 x + 36}{2 x - 14}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{- 4 x - 12}{x - 7} \cdot \frac{2 x - 14}{x^{2} - 12 x + 36}$

Step 2

Simplify with factoring out.

Factor $4$ out of $- 4 x - 12$ .

Factor $4$ out of $- 4 x$ .

$\frac{4 (- x) - 12}{x - 7} \cdot \frac{2 x - 14}{x^{2} - 12 x + 36}$

Factor $4$ out of $- 12$ .

$\frac{4 (- x) + 4 (- 3)}{x - 7} \cdot \frac{2 x - 14}{x^{2} - 12 x + 36}$

Factor $4$ out of $4 (- x) + 4 (- 3)$ .

$\frac{4 (- x - 3)}{x - 7} \cdot \frac{2 x - 14}{x^{2} - 12 x + 36}$

Factor $2$ out of $2 x - 14$ .

Factor $2$ out of $2 x$ .

$\frac{4 (- x - 3)}{x - 7} \cdot \frac{2 (x) - 14}{x^{2} - 12 x + 36}$

Factor $2$ out of $- 14$ .

$\frac{4 (- x - 3)}{x - 7} \cdot \frac{2 x + 2 \cdot - 7}{x^{2} - 12 x + 36}$

Factor $2$ out of $2 x + 2 \cdot - 7$ .

$\frac{4 (- x - 3)}{x - 7} \cdot \frac{2 (x - 7)}{x^{2} - 12 x + 36}$

Step 3

Factor using the perfect square rule.

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

$\frac{4 (- x - 3)}{x - 7} \cdot \frac{2 (x - 7)}{x^{2} - 12 x + 6^{2}}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$12 x = 2 \cdot x \cdot 6$

Rewrite the polynomial.

$\frac{4 (- x - 3)}{x - 7} \cdot \frac{2 (x - 7)}{x^{2} - 2 \cdot x \cdot 6 + 6^{2}}$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 6$ .

$\frac{4 (- x - 3)}{x - 7} \cdot \frac{2 (x - 7)}{{(x - 6)}^{2}}$

Step 4

Simplify terms.

Cancel the common factor of $x - 7$ .

Factor $x - 7$ out of $2 (x - 7)$ .

$\frac{4 (- x - 3)}{x - 7} \cdot \frac{(x - 7) \cdot 2}{{(x - 6)}^{2}}$

Cancel the common factor.

$\frac{4 (- x - 3)}{x - 7} \cdot \frac{(x - 7) \cdot 2}{{(x - 6)}^{2}}$

Rewrite the expression.

$4 (- x - 3) \frac{2}{{(x - 6)}^{2}}$

Combine $\frac{2}{{(x - 6)}^{2}}$ and $4$ .

$\frac{2 \cdot 4}{{(x - 6)}^{2}} (- x - 3)$

Multiply $2$ by $4$ .

$\frac{8}{{(x - 6)}^{2}} (- x - 3)$

Multiply $\frac{8}{{(x - 6)}^{2}}$ by $- x - 3$ .

$\frac{8 (- x - 3)}{{(x - 6)}^{2}}$

Factor $- 1$ out of $- x$ .

$\frac{8 (- (x) - 3)}{{(x - 6)}^{2}}$

Rewrite $- 3$ as $- 1 (3)$ .

$\frac{8 (- (x) - 1 (3))}{{(x - 6)}^{2}}$

Factor $- 1$ out of $- (x) - 1 (3)$ .

$\frac{8 (- (x + 3))}{{(x - 6)}^{2}}$

Simplify the expression.

Rewrite $- (x + 3)$ as $- 1 (x + 3)$ .

$\frac{8 (- 1 (x + 3))}{{(x - 6)}^{2}}$

Move the negative in front of the fraction.

$- \frac{8 (x + 3)}{{(x - 6)}^{2}}$

Simplify ((-4x-12)/(x-7))÷((x^2-12x+36)/(2x-14))

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