Simplify ((1-h^2)/(2h^2-10h-12))÷((2h-2)/6)

$\frac{1 - h^{2}}{2 h^{2} - 10 h - 12} \div \frac{2 h - 2}{6}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{1 - h^{2}}{2 h^{2} - 10 h - 12} \cdot \frac{6}{2 h - 2}$

Step 2

Simplify the numerator.

Rewrite $1$ as $1^{2}$ .

$\frac{1^{2} - h^{2}}{2 h^{2} - 10 h - 12} \cdot \frac{6}{2 h - 2}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = h$ .

$\frac{(1 + h) (1 - h)}{2 h^{2} - 10 h - 12} \cdot \frac{6}{2 h - 2}$

Step 3

Simplify the denominator.

Factor $2$ out of $2 h^{2} - 10 h - 12$ .

Factor $2$ out of $2 h^{2}$ .

$\frac{(1 + h) (1 - h)}{2 (h^{2}) - 10 h - 12} \cdot \frac{6}{2 h - 2}$

Factor $2$ out of $- 10 h$ .

$\frac{(1 + h) (1 - h)}{2 (h^{2}) + 2 (- 5 h) - 12} \cdot \frac{6}{2 h - 2}$

Factor $2$ out of $- 12$ .

$\frac{(1 + h) (1 - h)}{2 h^{2} + 2 (- 5 h) + 2 \cdot - 6} \cdot \frac{6}{2 h - 2}$

Factor $2$ out of $2 h^{2} + 2 (- 5 h)$ .

$\frac{(1 + h) (1 - h)}{2 (h^{2} - 5 h) + 2 \cdot - 6} \cdot \frac{6}{2 h - 2}$

Factor $2$ out of $2 (h^{2} - 5 h) + 2 \cdot - 6$ .

$\frac{(1 + h) (1 - h)}{2 (h^{2} - 5 h - 6)} \cdot \frac{6}{2 h - 2}$

Factor $h^{2} - 5 h - 6$ 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 $- 6$ and whose sum is $- 5$ .

$- 6, 1$

Write the factored form using these integers.

$\frac{(1 + h) (1 - h)}{2 ((h - 6) (h + 1))} \cdot \frac{6}{2 h - 2}$

$\frac{(1 + h) (1 - h)}{2 (h - 6) (h + 1)} \cdot \frac{6}{2 h - 2}$

Step 4

Simplify terms.

Cancel the common factor of $2$ .

Factor $2$ out of $2 (h - 6) (h + 1)$ .

$\frac{(1 + h) (1 - h)}{2 ((h - 6) (h + 1))} \cdot \frac{6}{2 h - 2}$

Factor $2$ out of $6$ .

$\frac{(1 + h) (1 - h)}{2 ((h - 6) (h + 1))} \cdot \frac{2 (3)}{2 h - 2}$

Cancel the common factor.

$\frac{(1 + h) (1 - h)}{2 ((h - 6) (h + 1))} \cdot \frac{2 \cdot 3}{2 h - 2}$

Rewrite the expression.

$\frac{(1 + h) (1 - h)}{(h - 6) (h + 1)} \cdot \frac{3}{2 h - 2}$

Multiply $\frac{(1 + h) (1 - h)}{(h - 6) (h + 1)}$ by $\frac{3}{2 h - 2}$ .

$\frac{(1 + h) (1 - h) \cdot 3}{(h - 6) (h + 1) (2 h - 2)}$

Cancel the common factor of $1 + h$ and $h + 1$ .

Reorder terms.

$\frac{(h + 1) (1 - h) \cdot 3}{(h - 6) (h + 1) (2 h - 2)}$

Cancel the common factor.

$\frac{(h + 1) (1 - h) \cdot 3}{(h - 6) (h + 1) (2 h - 2)}$

Rewrite the expression.

$\frac{(1 - h) \cdot 3}{(h - 6) (2 h - 2)}$

Step 5

Factor $2$ out of $2 h - 2$ .

Factor $2$ out of $2 h$ .

$\frac{(1 - h) \cdot 3}{(h - 6) (2 (h) - 2)}$

Factor $2$ out of $- 2$ .

$\frac{(1 - h) \cdot 3}{(h - 6) (2 (h) + 2 (- 1))}$

Factor $2$ out of $2 (h) + 2 (- 1)$ .

$\frac{(1 - h) \cdot 3}{(h - 6) (2 (h - 1))}$

$\frac{(1 - h) \cdot 3}{(h - 6) \cdot 2 (h - 1)}$

Step 6

Cancel the common factor of $1 - h$ and $h - 1$ .

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

$\frac{(- 1 (- 1) - h) \cdot 3}{(h - 6) \cdot 2 (h - 1)}$

Factor $- 1$ out of $- h$ .

$\frac{(- 1 (- 1) - (h)) \cdot 3}{(h - 6) \cdot 2 (h - 1)}$

Factor $- 1$ out of $- 1 (- 1) - (h)$ .

$\frac{- 1 (- 1 + h) \cdot 3}{(h - 6) \cdot 2 (h - 1)}$

Reorder terms.

$\frac{- 1 (h - 1) \cdot 3}{(h - 6) \cdot 2 (h - 1)}$

Cancel the common factor.

$\frac{- 1 (h - 1) \cdot 3}{(h - 6) \cdot 2 (h - 1)}$

Rewrite the expression.

$\frac{- 1 \cdot 3}{(h - 6) \cdot 2}$

Step 7

Simplify the expression.

Multiply $- 1$ by $3$ .

$\frac{- 3}{(h - 6) \cdot 2}$

Move $2$ to the left of $h - 6$ .

$\frac{- 3}{2 \cdot (h - 6)}$

Move the negative in front of the fraction.

$- \frac{3}{2 (h - 6)}$

Simplify ((1-h^2)/(2h^2-10h-12))÷((2h-2)/6)

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