Algebra Examples

$\frac{6 - x}{x^{2} - x - 30} \cdot \frac{x^{2} - 100}{2 x^{2} - 200} \div \frac{x + 10}{x + 5}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{6 - x}{x^{2} - x - 30} \cdot \frac{x^{2} - 100}{2 x^{2} - 200} \frac{x + 5}{x + 10}$

Step 2

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

$- 6, 5$

Step 2.2

Write the factored form using these integers.

$\frac{6 - x}{(x - 6) (x + 5)} \cdot \frac{x^{2} - 100}{2 x^{2} - 200} \frac{x + 5}{x + 10}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite $100$ as $10^{2}$ .

$\frac{6 - x}{(x - 6) (x + 5)} \cdot \frac{x^{2} - 10^{2}}{2 x^{2} - 200} \frac{x + 5}{x + 10}$

Step 3.2

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 $b = 10$ .

$\frac{6 - x}{(x - 6) (x + 5)} \cdot \frac{(x + 10) (x - 10)}{2 x^{2} - 200} \frac{x + 5}{x + 10}$

Step 4

Simplify the denominator.

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Step 4.1

Factor $2$ out of $2 x^{2} - 200$ .

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Step 4.1.1

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

$\frac{6 - x}{(x - 6) (x + 5)} \cdot \frac{(x + 10) (x - 10)}{2 (x^{2}) - 200} \frac{x + 5}{x + 10}$

Step 4.1.2

Factor $2$ out of $- 200$ .

$\frac{6 - x}{(x - 6) (x + 5)} \cdot \frac{(x + 10) (x - 10)}{2 x^{2} + 2 \cdot - 100} \frac{x + 5}{x + 10}$

Step 4.1.3

Factor $2$ out of $2 x^{2} + 2 \cdot - 100$ .

$\frac{6 - x}{(x - 6) (x + 5)} \cdot \frac{(x + 10) (x - 10)}{2 (x^{2} - 100)} \frac{x + 5}{x + 10}$

Step 4.2

Rewrite $100$ as $10^{2}$ .

$\frac{6 - x}{(x - 6) (x + 5)} \cdot \frac{(x + 10) (x - 10)}{2 (x^{2} - 10^{2})} \frac{x + 5}{x + 10}$

Step 4.3

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 $b = 10$ .

$\frac{6 - x}{(x - 6) (x + 5)} \cdot \frac{(x + 10) (x - 10)}{2 (x + 10) (x - 10)} \frac{x + 5}{x + 10}$

Step 5

Simplify terms.

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Step 5.1

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

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Step 5.1.1

Rewrite $6$ as $- 1 (- 6)$ .

$\frac{- 1 (- 6) - x}{(x - 6) (x + 5)} \cdot \frac{(x + 10) (x - 10)}{2 (x + 10) (x - 10)} \frac{x + 5}{x + 10}$

Step 5.1.2

Factor $- 1$ out of $- x$ .

$\frac{- 1 (- 6) - (x)}{(x - 6) (x + 5)} \cdot \frac{(x + 10) (x - 10)}{2 (x + 10) (x - 10)} \frac{x + 5}{x + 10}$

Step 5.1.3

Factor $- 1$ out of $- 1 (- 6) - (x)$ .

$\frac{- 1 (- 6 + x)}{(x - 6) (x + 5)} \cdot \frac{(x + 10) (x - 10)}{2 (x + 10) (x - 10)} \frac{x + 5}{x + 10}$

Step 5.1.4

Reorder terms.

$\frac{- 1 (x - 6)}{(x - 6) (x + 5)} \cdot \frac{(x + 10) (x - 10)}{2 (x + 10) (x - 10)} \frac{x + 5}{x + 10}$

Step 5.1.5

Cancel the common factor.

$\frac{- 1 (x - 6)}{(x - 6) (x + 5)} \cdot \frac{(x + 10) (x - 10)}{2 (x + 10) (x - 10)} \frac{x + 5}{x + 10}$

Step 5.1.6

Rewrite the expression.

$\frac{- 1}{x + 5} \cdot \frac{(x + 10) (x - 10)}{2 (x + 10) (x - 10)} \frac{x + 5}{x + 10}$

Step 5.2

Move the negative in front of the fraction.

$- \frac{1}{x + 5} \cdot \frac{(x + 10) (x - 10)}{2 (x + 10) (x - 10)} \frac{x + 5}{x + 10}$

Step 5.3

Cancel the common factor of $x + 10$ .

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Step 5.3.1

Cancel the common factor.

$- \frac{1}{x + 5} \cdot \frac{(x + 10) (x - 10)}{2 (x + 10) (x - 10)} \frac{x + 5}{x + 10}$

Step 5.3.2

Rewrite the expression.

$- \frac{1}{x + 5} \cdot \frac{x - 10}{2 (x - 10)} \frac{x + 5}{x + 10}$

Step 5.4

Cancel the common factor of $x - 10$ .

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Step 5.4.1

Cancel the common factor.

$- \frac{1}{x + 5} \cdot \frac{x - 10}{2 (x - 10)} \frac{x + 5}{x + 10}$

Step 5.4.2

Rewrite the expression.

$- \frac{1}{x + 5} \cdot \frac{1}{2} \frac{x + 5}{x + 10}$

Step 5.5

Multiply $\frac{1}{2}$ by $\frac{1}{x + 5}$ .

$- \frac{1}{2 (x + 5)} \cdot \frac{x + 5}{x + 10}$

Step 5.6

Cancel the common factor of $x + 5$ .

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Step 5.6.1

Move the leading negative in $- \frac{1}{2 (x + 5)}$ into the numerator.

$\frac{- 1}{2 (x + 5)} \cdot \frac{x + 5}{x + 10}$

Step 5.6.2

Factor $x + 5$ out of $2 (x + 5)$ .

$\frac{- 1}{(x + 5) \cdot 2} \cdot \frac{x + 5}{x + 10}$

Step 5.6.3

Cancel the common factor.

$\frac{- 1}{(x + 5) \cdot 2} \cdot \frac{x + 5}{x + 10}$

Step 5.6.4

Rewrite the expression.

$\frac{- 1}{2} \cdot \frac{1}{x + 10}$

Step 5.7

Multiply $\frac{- 1}{2}$ by $\frac{1}{x + 10}$ .

$\frac{- 1}{2 (x + 10)}$

Step 5.8

Move the negative in front of the fraction.

$- \frac{1}{2 (x + 10)}$