Algebra Examples

$\frac{x + 8}{6 x + 60} \div \frac{x^{2} - 64}{x^{2} + 2 x - 80}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x + 8}{6 x + 60} \cdot \frac{x^{2} + 2 x - 80}{x^{2} - 64}$

Step 2

Factor $6$ out of $6 x + 60$ .

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Step 2.1

Factor $6$ out of $6 x$ .

$\frac{x + 8}{6 (x) + 60} \cdot \frac{x^{2} + 2 x - 80}{x^{2} - 64}$

Step 2.2

Factor $6$ out of $60$ .

$\frac{x + 8}{6 x + 6 \cdot 10} \cdot \frac{x^{2} + 2 x - 80}{x^{2} - 64}$

Step 2.3

Factor $6$ out of $6 x + 6 \cdot 10$ .

$\frac{x + 8}{6 (x + 10)} \cdot \frac{x^{2} + 2 x - 80}{x^{2} - 64}$

Step 3

Factor $x^{2} + 2 x - 80$ 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 $- 80$ and whose sum is $2$ .

$- 8, 10$

Step 3.2

Write the factored form using these integers.

$\frac{x + 8}{6 (x + 10)} \cdot \frac{(x - 8) (x + 10)}{x^{2} - 64}$

Step 4

Simplify the denominator.

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Step 4.1

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

$\frac{x + 8}{6 (x + 10)} \cdot \frac{(x - 8) (x + 10)}{x^{2} - 8^{2}}$

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 $b = 8$ .

$\frac{x + 8}{6 (x + 10)} \cdot \frac{(x - 8) (x + 10)}{(x + 8) (x - 8)}$

Step 5

Cancel the common factor of $x + 8$ .

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Step 5.1

Cancel the common factor.

$\frac{x + 8}{6 (x + 10)} \cdot \frac{(x - 8) (x + 10)}{(x + 8) (x - 8)}$

Step 5.2

Rewrite the expression.

$\frac{1}{6 (x + 10)} \cdot \frac{(x - 8) (x + 10)}{x - 8}$

Step 6

Cancel the common factor of $x + 10$ .

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Step 6.1

Factor $x + 10$ out of $6 (x + 10)$ .

$\frac{1}{(x + 10) \cdot 6} \cdot \frac{(x - 8) (x + 10)}{x - 8}$

Step 6.2

Factor $x + 10$ out of $(x - 8) (x + 10)$ .

$\frac{1}{(x + 10) \cdot 6} \cdot \frac{(x + 10) (x - 8)}{x - 8}$

Step 6.3

Cancel the common factor.

$\frac{1}{(x + 10) \cdot 6} \cdot \frac{(x + 10) (x - 8)}{x - 8}$

Step 6.4

Rewrite the expression.

$\frac{1}{6} \cdot \frac{x - 8}{x - 8}$

Step 7

Multiply $\frac{1}{6}$ by $\frac{x - 8}{x - 8}$ .

$\frac{x - 8}{6 (x - 8)}$

Step 8

Cancel the common factor of $x - 8$ .

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Step 8.1

Cancel the common factor.

$\frac{x - 8}{6 (x - 8)}$

Step 8.2

Rewrite the expression.

$\frac{1}{6}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{6}$

Decimal Form:

$0.1\overline{6}$