Algebra Examples

$\frac{x^{2} - 4}{5 x + 30} \div \frac{x^{2} + 11 x + 18}{3 x + 27}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 4}{5 x + 30} \cdot \frac{3 x + 27}{x^{2} + 11 x + 18}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $4$ as $2^{2}$ .

$\frac{x^{2} - 2^{2}}{5 x + 30} \cdot \frac{3 x + 27}{x^{2} + 11 x + 18}$

Step 2.2

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 $b = 2$ .

$\frac{(x + 2) (x - 2)}{5 x + 30} \cdot \frac{3 x + 27}{x^{2} + 11 x + 18}$

Step 3

Factor $5$ out of $5 x + 30$ .

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Step 3.1

Factor $5$ out of $5 x$ .

$\frac{(x + 2) (x - 2)}{5 (x) + 30} \cdot \frac{3 x + 27}{x^{2} + 11 x + 18}$

Step 3.2

Factor $5$ out of $30$ .

$\frac{(x + 2) (x - 2)}{5 x + 5 \cdot 6} \cdot \frac{3 x + 27}{x^{2} + 11 x + 18}$

Step 3.3

Factor $5$ out of $5 x + 5 \cdot 6$ .

$\frac{(x + 2) (x - 2)}{5 (x + 6)} \cdot \frac{3 x + 27}{x^{2} + 11 x + 18}$

Step 4

Factor $3$ out of $3 x + 27$ .

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Step 4.1

Factor $3$ out of $3 x$ .

$\frac{(x + 2) (x - 2)}{5 (x + 6)} \cdot \frac{3 (x) + 27}{x^{2} + 11 x + 18}$

Step 4.2

Factor $3$ out of $27$ .

$\frac{(x + 2) (x - 2)}{5 (x + 6)} \cdot \frac{3 x + 3 \cdot 9}{x^{2} + 11 x + 18}$

Step 4.3

Factor $3$ out of $3 x + 3 \cdot 9$ .

$\frac{(x + 2) (x - 2)}{5 (x + 6)} \cdot \frac{3 (x + 9)}{x^{2} + 11 x + 18}$

Step 5

Factor $x^{2} + 11 x + 18$ using the AC method.

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Step 5.1

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 $11$ .

$2, 9$

Step 5.2

Write the factored form using these integers.

$\frac{(x + 2) (x - 2)}{5 (x + 6)} \cdot \frac{3 (x + 9)}{(x + 2) (x + 9)}$

Step 6

Cancel the common factor of $x + 2$ .

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Step 6.1

Cancel the common factor.

$\frac{(x + 2) (x - 2)}{5 (x + 6)} \cdot \frac{3 (x + 9)}{(x + 2) (x + 9)}$

Step 6.2

Rewrite the expression.

$\frac{x - 2}{5 (x + 6)} \cdot \frac{3 (x + 9)}{x + 9}$

Step 7

Multiply $\frac{x - 2}{5 (x + 6)}$ by $\frac{3 (x + 9)}{x + 9}$ .

$\frac{(x - 2) (3 (x + 9))}{5 (x + 6) (x + 9)}$

Step 8

Cancel the common factor of $x + 9$ .

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Step 8.1

Cancel the common factor.

$\frac{(x - 2) (3 (x + 9))}{5 (x + 6) (x + 9)}$

Step 8.2

Rewrite the expression.

$\frac{(x - 2) \cdot (3)}{5 (x + 6)}$

Step 9

Move $3$ to the left of $x - 2$ .

$\frac{3 (x - 2)}{5 (x + 6)}$