Simplify ((x^2-1)/(x^2-5x-6))÷((5x+25)/(x-6))

$\frac{x^{2} - 1}{x^{2} - 5 x - 6} \div \frac{5 x + 25}{x - 6}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 1}{x^{2} - 5 x - 6} \cdot \frac{x - 6}{5 x + 25}$

Step 2

Simplify the numerator.

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

$\frac{x^{2} - 1^{2}}{x^{2} - 5 x - 6} \cdot \frac{x - 6}{5 x + 25}$

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 $b = 1$ .

$\frac{(x + 1) (x - 1)}{x^{2} - 5 x - 6} \cdot \frac{x - 6}{5 x + 25}$

Step 3

Factor $x^{2} - 5 x - 6$ 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.

$\frac{(x + 1) (x - 1)}{(x - 6) (x + 1)} \cdot \frac{x - 6}{5 x + 25}$

Step 4

Simplify terms.

Cancel the common factor of $x - 6$ .

Cancel the common factor.

$\frac{(x + 1) (x - 1)}{(x - 6) (x + 1)} \cdot \frac{x - 6}{5 x + 25}$

Rewrite the expression.

$\frac{(x + 1) (x - 1)}{x + 1} \cdot \frac{1}{5 x + 25}$

Multiply $\frac{(x + 1) (x - 1)}{x + 1}$ by $\frac{1}{5 x + 25}$ .

$\frac{(x + 1) (x - 1)}{(x + 1) (5 x + 25)}$

Cancel the common factor of $x + 1$ .

Cancel the common factor.

$\frac{(x + 1) (x - 1)}{(x + 1) (5 x + 25)}$

Rewrite the expression.

$\frac{x - 1}{5 x + 25}$

Factor $5$ out of $5 x + 25$ .

Factor $5$ out of $5 x$ .

$\frac{x - 1}{5 (x) + 25}$

Factor $5$ out of $25$ .

$\frac{x - 1}{5 x + 5 \cdot 5}$

Factor $5$ out of $5 x + 5 \cdot 5$ .

$\frac{x - 1}{5 (x + 5)}$

Simplify ((x^2-1)/(x^2-5x-6))÷((5x+25)/(x-6))

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