Algebra Examples

$\frac{x - 4}{2 x - 18} \cdot \frac{x^{2} - 7 x - 18}{x^{2} - 16}$

Step 1

Factor $2$ out of $2 x - 18$ .

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Step 1.1

Factor $2$ out of $2 x$ .

$\frac{x - 4}{2 (x) - 18} \cdot \frac{x^{2} - 7 x - 18}{x^{2} - 16}$

Step 1.2

Factor $2$ out of $- 18$ .

$\frac{x - 4}{2 x + 2 \cdot - 9} \cdot \frac{x^{2} - 7 x - 18}{x^{2} - 16}$

Step 1.3

Factor $2$ out of $2 x + 2 \cdot - 9$ .

$\frac{x - 4}{2 (x - 9)} \cdot \frac{x^{2} - 7 x - 18}{x^{2} - 16}$

Step 2

Factor $x^{2} - 7 x - 18$ 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 $- 7$ .

$- 9, 2$

Step 2.2

Write the factored form using these integers.

$\frac{x - 4}{2 (x - 9)} \cdot \frac{(x - 9) (x + 2)}{x^{2} - 16}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $16$ as $4^{2}$ .

$\frac{x - 4}{2 (x - 9)} \cdot \frac{(x - 9) (x + 2)}{x^{2} - 4^{2}}$

Step 3.2

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 $b = 4$ .

$\frac{x - 4}{2 (x - 9)} \cdot \frac{(x - 9) (x + 2)}{(x + 4) (x - 4)}$

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $x - 4$ .

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Step 4.1.1

Factor $x - 4$ out of $(x + 4) (x - 4)$ .

$\frac{x - 4}{2 (x - 9)} \cdot \frac{(x - 9) (x + 2)}{(x - 4) (x + 4)}$

Step 4.1.2

Cancel the common factor.

$\frac{x - 4}{2 (x - 9)} \cdot \frac{(x - 9) (x + 2)}{(x - 4) (x + 4)}$

Step 4.1.3

Rewrite the expression.

$\frac{1}{2 (x - 9)} \cdot \frac{(x - 9) (x + 2)}{x + 4}$

Step 4.2

Cancel the common factor of $x - 9$ .

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Step 4.2.1

Factor $x - 9$ out of $2 (x - 9)$ .

$\frac{1}{(x - 9) \cdot 2} \cdot \frac{(x - 9) (x + 2)}{x + 4}$

Step 4.2.2

Cancel the common factor.

$\frac{1}{(x - 9) \cdot 2} \cdot \frac{(x - 9) (x + 2)}{x + 4}$

Step 4.2.3

Rewrite the expression.

$\frac{1}{2} \cdot \frac{x + 2}{x + 4}$

Step 4.3

Multiply $\frac{1}{2}$ by $\frac{x + 2}{x + 4}$ .

$\frac{x + 2}{2 (x + 4)}$