Algebra Examples

$\frac{x^{2} - 4 x - 45}{x^{2} - x - 30} \cdot \frac{x^{2} - 36}{36 - x^{2}} \div \frac{x - 9}{6 x}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 4 x - 45}{x^{2} - x - 30} \cdot \frac{x^{2} - 36}{36 - x^{2}} \frac{6 x}{x - 9}$

Step 2

Factor $x^{2} - 4 x - 45$ 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 $- 45$ and whose sum is $- 4$ .

$- 9, 5$

Step 2.2

Write the factored form using these integers.

$\frac{(x - 9) (x + 5)}{x^{2} - x - 30} \cdot \frac{x^{2} - 36}{36 - x^{2}} \frac{6 x}{x - 9}$

Step 3

Factor $x^{2} - x - 30$ 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 $- 30$ and whose sum is $- 1$ .

$- 6, 5$

Step 3.2

Write the factored form using these integers.

$\frac{(x - 9) (x + 5)}{(x - 6) (x + 5)} \cdot \frac{x^{2} - 36}{36 - x^{2}} \frac{6 x}{x - 9}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite $36$ as $6^{2}$ .

$\frac{(x - 9) (x + 5)}{(x - 6) (x + 5)} \cdot \frac{x^{2} - 6^{2}}{36 - x^{2}} \frac{6 x}{x - 9}$

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 $b = 6$ .

$\frac{(x - 9) (x + 5)}{(x - 6) (x + 5)} \cdot \frac{(x + 6) (x - 6)}{36 - x^{2}} \frac{6 x}{x - 9}$

Step 5

Simplify the denominator.

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Step 5.1

Rewrite $36$ as $6^{2}$ .

$\frac{(x - 9) (x + 5)}{(x - 6) (x + 5)} \cdot \frac{(x + 6) (x - 6)}{6^{2} - x^{2}} \frac{6 x}{x - 9}$

Step 5.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 6$ and $b = x$ .

$\frac{(x - 9) (x + 5)}{(x - 6) (x + 5)} \cdot \frac{(x + 6) (x - 6)}{(6 + x) (6 - x)} \frac{6 x}{x - 9}$

Step 6

Simplify terms.

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Step 6.1

Cancel the common factor of $x - 6$ .

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Step 6.1.1

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

$\frac{(x - 9) (x + 5)}{(x - 6) (x + 5)} \cdot \frac{(x - 6) (x + 6)}{(6 + x) (6 - x)} \frac{6 x}{x - 9}$

Step 6.1.2

Cancel the common factor.

$\frac{(x - 9) (x + 5)}{(x - 6) (x + 5)} \cdot \frac{(x - 6) (x + 6)}{(6 + x) (6 - x)} \frac{6 x}{x - 9}$

Step 6.1.3

Rewrite the expression.

$\frac{(x - 9) (x + 5)}{x + 5} \cdot \frac{x + 6}{(6 + x) (6 - x)} \frac{6 x}{x - 9}$

Step 6.2

Multiply $\frac{(x - 9) (x + 5)}{x + 5}$ by $\frac{x + 6}{(6 + x) (6 - x)}$ .

$\frac{(x - 9) (x + 5) (x + 6)}{(x + 5) ((6 + x) (6 - x))} \cdot \frac{6 x}{x - 9}$

Step 6.3

Cancel the common factor of $x - 9$ .

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Step 6.3.1

Factor $x - 9$ out of $(x - 9) (x + 5) (x + 6)$ .

$\frac{(x - 9) ((x + 5) (x + 6))}{(x + 5) ((6 + x) (6 - x))} \cdot \frac{6 x}{x - 9}$

Step 6.3.2

Cancel the common factor.

$\frac{(x - 9) ((x + 5) (x + 6))}{(x + 5) ((6 + x) (6 - x))} \cdot \frac{6 x}{x - 9}$

Step 6.3.3

Rewrite the expression.

$\frac{(x + 5) (x + 6)}{(x + 5) ((6 + x) (6 - x))} (6 x)$

Step 6.4

Combine $6$ and $\frac{(x + 5) (x + 6)}{(x + 5) ((6 + x) (6 - x))}$ .

$\frac{6 ((x + 5) (x + 6))}{(x + 5) ((6 + x) (6 - x))} x$

Step 6.5

Combine $\frac{6 ((x + 5) (x + 6))}{(x + 5) ((6 + x) (6 - x))}$ and $x$ .

$\frac{6 ((x + 5) (x + 6)) x}{(x + 5) ((6 + x) (6 - x))}$

Step 6.6

Cancel the common factor of $x + 5$ .

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Step 6.6.1

Cancel the common factor.

$\frac{6 ((x + 5) (x + 6)) x}{(x + 5) ((6 + x) (6 - x))}$

Step 6.6.2

Rewrite the expression.

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

Step 6.7

Cancel the common factor of $x + 6$ and $6 + x$ .

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Step 6.7.1

Reorder terms.

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

Step 6.7.2

Cancel the common factor.

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

Step 6.7.3

Rewrite the expression.

$\frac{6 x}{6 - x}$