Divide ((x^2+9x+18)/(x^2-9))/((x+6)/(x-6))

$\frac{\frac{x^{2} + 9 x + 18}{x^{2} - 9}}{\frac{x + 6}{x - 6}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{x^{2} + 9 x + 18}{x^{2} - 9} \cdot \frac{x - 6}{x + 6}$

Step 2

Factor $x^{2} + 9 x + 18$ 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 $18$ and whose sum is $9$ .

$3, 6$

Write the factored form using these integers.

$\frac{(x + 3) (x + 6)}{x^{2} - 9} \cdot \frac{x - 6}{x + 6}$

Step 3

Simplify the denominator.

Rewrite $9$ as $3^{2}$ .

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

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 $b = 3$ .

$\frac{(x + 3) (x + 6)}{(x + 3) (x - 3)} \cdot \frac{x - 6}{x + 6}$

Step 4

Cancel the common factor of $x + 6$ .

Factor $x + 6$ out of $(x + 3) (x + 6)$ .

$\frac{(x + 6) (x + 3)}{(x + 3) (x - 3)} \cdot \frac{x - 6}{x + 6}$

Cancel the common factor.

$\frac{(x + 6) (x + 3)}{(x + 3) (x - 3)} \cdot \frac{x - 6}{x + 6}$

Rewrite the expression.

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

Step 5

Cancel the common factor of $x + 3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Step 6

Multiply $\frac{1}{x - 3}$ by $x - 6$ .

$\frac{x - 6}{x - 3}$

Divide ((x^2+9x+18)/(x^2-9))/((x+6)/(x-6))

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