Algebra Examples

$\frac{x^{2} - 64}{x - 8} \cdot \frac{x + 5}{x^{2} + 13 x + 40}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $64$ as $8^{2}$ .

$\frac{x^{2} - 8^{2}}{x - 8} \cdot \frac{x + 5}{x^{2} + 13 x + 40}$

Step 1.2

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 $b = 8$ .

$\frac{(x + 8) (x - 8)}{x - 8} \cdot \frac{x + 5}{x^{2} + 13 x + 40}$

Step 2

Factor $x^{2} + 13 x + 40$ 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 $40$ and whose sum is $13$ .

$5, 8$

Step 2.2

Write the factored form using these integers.

$\frac{(x + 8) (x - 8)}{x - 8} \cdot \frac{x + 5}{(x + 5) (x + 8)}$

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $x + 8$ .

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Step 3.1.1

Factor $x + 8$ out of $(x + 5) (x + 8)$ .

$\frac{(x + 8) (x - 8)}{x - 8} \cdot \frac{x + 5}{(x + 8) (x + 5)}$

Step 3.1.2

Cancel the common factor.

$\frac{(x + 8) (x - 8)}{x - 8} \cdot \frac{x + 5}{(x + 8) (x + 5)}$

Step 3.1.3

Rewrite the expression.

$\frac{x - 8}{x - 8} \cdot \frac{x + 5}{x + 5}$

Step 3.2

Multiply $\frac{x - 8}{x - 8}$ by $\frac{x + 5}{x + 5}$ .

$\frac{(x - 8) (x + 5)}{(x - 8) (x + 5)}$

Step 3.3

Cancel the common factor of $x - 8$ .

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Step 3.3.1

Cancel the common factor.

$\frac{(x - 8) (x + 5)}{(x - 8) (x + 5)}$

Step 3.3.2

Rewrite the expression.

$\frac{x + 5}{x + 5}$

Step 3.4

Cancel the common factor of $x + 5$ .

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Step 3.4.1

Cancel the common factor.

$\frac{x + 5}{x + 5}$

Step 3.4.2

Rewrite the expression.

$1$