Simplify ((x^2-x-6)/(x^2-9))÷((3x+6)/(x+3))

$\frac{x^{2} - x - 6}{x^{2} - 9} \div \frac{3 x + 6}{x + 3}$

To divide by a fraction, multiply by its reciprocal.

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

Factor $x^{2} - x - 6$ 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 $- 1$ .

$- 3, 2$

Write the factored form using these integers.

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

Simplify the denominator.

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

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

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 $b = 3$ .

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

Simplify terms.

Cancel the common factor of $x + 3$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Cancel the common factor of $x - 3$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{x + 2}{3 x + 6}$

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

Factor $3$ out of $3 x$ .

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

Factor $3$ out of $6$ .

$\frac{x + 2}{3 x + 3 \cdot 2}$

Factor $3$ out of $3 x + 3 \cdot 2$ .

$\frac{x + 2}{3 (x + 2)}$

Cancel the common factor of $x + 2$ .

Cancel the common factor.

$\frac{x + 2}{3 (x + 2)}$

Rewrite the expression.

$\frac{1}{3}$

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{3}$

Decimal Form:

$0.\overline{3}$

Simplify ((x^2-x-6)/(x^2-9))÷((3x+6)/(x+3))

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