Algebra Examples

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

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

Factor $x^{2} - 9 x + 18$ using the AC method.

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Step 2.1

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 $- 9$ .

$- 6, - 3$

Step 2.2

Write the factored form using these integers.

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

Step 3

Factor $x^{2} + 9 x + 18$ using the AC method.

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Step 3.1

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 $9$ .

$3, 6$

Step 3.2

Write the factored form using these integers.

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

Step 4

Simplify the numerator.

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Step 4.1

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

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

Step 4.2

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 $b = 3$ .

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

Step 5

Simplify the denominator.

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Step 5.1

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

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

Step 5.2

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 $b = 6$ .

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

Step 6

Simplify terms.

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Step 6.1

Cancel the common factor of $x - 6$ .

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Step 6.1.1

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

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

Step 6.1.2

Cancel the common factor.

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

Step 6.1.3

Rewrite the expression.

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

Step 6.2

Cancel the common factor of $x + 3$ .

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Step 6.2.1

Cancel the common factor.

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

Step 6.2.2

Rewrite the expression.

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

Step 6.3

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

Add $1$ and $1$ .

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

Step 11

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

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

Step 12

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

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

Step 13

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

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

Step 14

Add $1$ and $1$ .

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