Divide ((x^2-9x+18)/(x^2+9x+18))÷((x^2-36)/(x^2-9))

$\frac{x^{2} - 9 x + 18}{x^{2} + 9 x + 18} \div \frac{x^{2} - 36}{x^{2} - 9}$

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 9 x + 18}{x^{2} + 9 x + 18} \cdot \frac{x^{2} - 9}{x^{2} - 36}$

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 $- 9$ .

$- 6, - 3$

Write the factored form using these integers.

$\frac{(x - 6) (x - 3)}{x^{2} + 9 x + 18} \cdot \frac{x^{2} - 9}{x^{2} - 36}$

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.

$\frac{(x - 6) (x - 3)}{(x + 3) (x + 6)} \cdot \frac{x^{2} - 9}{x^{2} - 36}$

Simplify the numerator.

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

$\frac{(x - 6) (x - 3)}{(x + 3) (x + 6)} \cdot \frac{x^{2} - 3^{2}}{x^{2} - 36}$

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 $b = 3$ .

$\frac{(x - 6) (x - 3)}{(x + 3) (x + 6)} \cdot \frac{(x + 3) (x - 3)}{x^{2} - 36}$

Simplify the denominator.

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

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

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 $b = 6$ .

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

Cancel the common factor of $x - 6$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x + 3$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Raise $x - 3$ to the power of $1$ .

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

Raise $x - 3$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $1$ and $1$ .

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

Raise $x + 6$ to the power of $1$ .

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

Raise $x + 6$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $1$ and $1$ .

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

Rewrite ${(x - 3)}^{2}$ as $(x - 3) (x - 3)$ .

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

Expand $(x - 3) (x - 3)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 3$ to the left of $x$ .

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

Multiply $- 3$ by $- 3$ .

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

Subtract $3 x$ from $- 3 x$ .

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

Split the fraction $\frac{x^{2} - 6 x + 9}{{(x + 6)}^{2}}$ into two fractions.

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

Split the fraction $\frac{x^{2} - 6 x}{{(x + 6)}^{2}}$ into two fractions.

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

Move the negative in front of the fraction.

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

Divide ((x^2-9x+18)/(x^2+9x+18))÷((x^2-36)/(x^2-9))

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