Pre-Algebra Examples

$\frac{x^{2} - 6 x}{x^{3} + 3 x^{2}} \div \frac{x^{2} + 3 x - 54}{x^{2} + 9 x}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 6 x}{x^{3} + 3 x^{2}} \cdot \frac{x^{2} + 9 x}{x^{2} + 3 x - 54}$

Step 2

Factor $x$ out of $x^{2} - 6 x$ .

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Step 2.1

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

$\frac{x \cdot x - 6 x}{x^{3} + 3 x^{2}} \cdot \frac{x^{2} + 9 x}{x^{2} + 3 x - 54}$

Step 2.2

Factor $x$ out of $- 6 x$ .

$\frac{x \cdot x + x \cdot - 6}{x^{3} + 3 x^{2}} \cdot \frac{x^{2} + 9 x}{x^{2} + 3 x - 54}$

Step 2.3

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

$\frac{x (x - 6)}{x^{3} + 3 x^{2}} \cdot \frac{x^{2} + 9 x}{x^{2} + 3 x - 54}$

Step 3

Factor $x^{2}$ out of $x^{3} + 3 x^{2}$ .

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Step 3.1

Factor $x^{2}$ out of $x^{3}$ .

$\frac{x (x - 6)}{x^{2} x + 3 x^{2}} \cdot \frac{x^{2} + 9 x}{x^{2} + 3 x - 54}$

Step 3.2

Factor $x^{2}$ out of $3 x^{2}$ .

$\frac{x (x - 6)}{x^{2} x + x^{2} \cdot 3} \cdot \frac{x^{2} + 9 x}{x^{2} + 3 x - 54}$

Step 3.3

Factor $x^{2}$ out of $x^{2} x + x^{2} \cdot 3$ .

$\frac{x (x - 6)}{x^{2} (x + 3)} \cdot \frac{x^{2} + 9 x}{x^{2} + 3 x - 54}$

Step 4

Cancel the common factors.

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Step 4.1

Factor $x$ out of $x^{2} (x + 3)$ .

$\frac{x (x - 6)}{x (x (x + 3))} \cdot \frac{x^{2} + 9 x}{x^{2} + 3 x - 54}$

Step 4.2

Cancel the common factor.

$\frac{x (x - 6)}{x (x (x + 3))} \cdot \frac{x^{2} + 9 x}{x^{2} + 3 x - 54}$

Step 4.3

Rewrite the expression.

$\frac{x - 6}{x (x + 3)} \cdot \frac{x^{2} + 9 x}{x^{2} + 3 x - 54}$

Step 5

Factor $x$ out of $x^{2} + 9 x$ .

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Step 5.1

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

$\frac{x - 6}{x (x + 3)} \cdot \frac{x \cdot x + 9 x}{x^{2} + 3 x - 54}$

Step 5.2

Factor $x$ out of $9 x$ .

$\frac{x - 6}{x (x + 3)} \cdot \frac{x \cdot x + x \cdot 9}{x^{2} + 3 x - 54}$

Step 5.3

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

$\frac{x - 6}{x (x + 3)} \cdot \frac{x (x + 9)}{x^{2} + 3 x - 54}$

Step 6

Factor $x^{2} + 3 x - 54$ using the AC method.

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Step 6.1

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 $- 54$ and whose sum is $3$ .

$- 6, 9$

Step 6.2

Write the factored form using these integers.

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

Step 7

Simplify terms.

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Step 7.1

Cancel the common factor of $x - 6$ .

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Step 7.1.1

Cancel the common factor.

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

Step 7.1.2

Rewrite the expression.

$\frac{1}{x (x + 3)} \cdot \frac{x (x + 9)}{x + 9}$

Step 7.2

Cancel the common factor of $x$ .

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Step 7.2.1

Cancel the common factor.

$\frac{1}{x (x + 3)} \cdot \frac{x (x + 9)}{x + 9}$

Step 7.2.2

Rewrite the expression.

$\frac{1}{x + 3} \cdot \frac{x + 9}{x + 9}$

Step 7.3

Multiply $\frac{1}{x + 3}$ by $\frac{x + 9}{x + 9}$ .

$\frac{x + 9}{(x + 3) (x + 9)}$

Step 7.4

Cancel the common factor of $x + 9$ .

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Step 7.4.1

Cancel the common factor.

$\frac{x + 9}{(x + 3) (x + 9)}$

Step 7.4.2

Rewrite the expression.

$\frac{1}{x + 3}$