Algebra Examples

$\frac{x^{2} + x}{x^{2} - 7 x - 8} \div \frac{x^{2}}{x^{2} - 8 x}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} + x}{x^{2} - 7 x - 8} \cdot \frac{x^{2} - 8 x}{x^{2}}$

Step 2

Factor $x$ out of $x^{2} + x$ .

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Step 2.1

Factor $x$ out of $x^{2}$ .

$\frac{x \cdot x + x}{x^{2} - 7 x - 8} \cdot \frac{x^{2} - 8 x}{x^{2}}$

Step 2.2

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

$\frac{x \cdot x + x^{1}}{x^{2} - 7 x - 8} \cdot \frac{x^{2} - 8 x}{x^{2}}$

Step 2.3

Factor $x$ out of $x^{1}$ .

$\frac{x \cdot x + x \cdot 1}{x^{2} - 7 x - 8} \cdot \frac{x^{2} - 8 x}{x^{2}}$

Step 2.4

Factor $x$ out of $x \cdot x + x \cdot 1$ .

$\frac{x (x + 1)}{x^{2} - 7 x - 8} \cdot \frac{x^{2} - 8 x}{x^{2}}$

Step 3

Factor $x^{2} - 7 x - 8$ 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 $- 8$ and whose sum is $- 7$ .

$- 8, 1$

Step 3.2

Write the factored form using these integers.

$\frac{x (x + 1)}{(x - 8) (x + 1)} \cdot \frac{x^{2} - 8 x}{x^{2}}$

Step 4

Cancel the common factor of $x$ .

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Step 4.1

Factor $x$ out of $x^{2}$ .

$\frac{x (x + 1)}{(x - 8) (x + 1)} \cdot \frac{x^{2} - 8 x}{x \cdot x}$

Step 4.2

Cancel the common factor.

$\frac{x (x + 1)}{(x - 8) (x + 1)} \cdot \frac{x^{2} - 8 x}{x \cdot x}$

Step 4.3

Rewrite the expression.

$\frac{x + 1}{(x - 8) (x + 1)} \cdot \frac{x^{2} - 8 x}{x}$

Step 5

Multiply $\frac{x + 1}{(x - 8) (x + 1)}$ by $\frac{x^{2} - 8 x}{x}$ .

$\frac{(x + 1) (x^{2} - 8 x)}{(x - 8) (x + 1) x}$

Step 6

Cancel the common factor of $x + 1$ .

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Step 6.1

Cancel the common factor.

$\frac{(x + 1) (x^{2} - 8 x)}{(x - 8) (x + 1) x}$

Step 6.2

Rewrite the expression.

$\frac{x^{2} - 8 x}{(x - 8) x}$

Step 7

Factor $x$ out of $x^{2} - 8 x$ .

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Step 7.1

Factor $x$ out of $x^{2}$ .

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

Step 7.2

Factor $x$ out of $- 8 x$ .

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

Step 7.3

Factor $x$ out of $x \cdot x + x \cdot - 8$ .

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

Step 8

Cancel the common factor of $x$ .

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Step 8.1

Cancel the common factor.

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

Step 8.2

Rewrite the expression.

$\frac{x - 8}{x - 8}$

Step 9

Cancel the common factor of $x - 8$ .

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Step 9.1

Cancel the common factor.

$\frac{x - 8}{x - 8}$

Step 9.2

Rewrite the expression.

$1$