Algebra Examples

$\frac{2 x^{2} - 12 x - 80}{- x - 2} \cdot \frac{4 - x^{2}}{x^{2} - 4} \cdot \frac{- x - 2}{x^{2} + 2 x - 8}$

Step 1

Factor $x^{2} + 2 x - 8$ using the AC method.

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Step 1.1

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 $2$ .

$- 2, 4$

Step 1.2

Write the factored form using these integers.

$\frac{2 x^{2} - 12 x - 80}{- x - 2} \cdot \frac{4 - x^{2}}{x^{2} - 4} \cdot \frac{- x - 2}{(x - 2) (x + 4)}$

Step 2

Simplify the numerator.

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Step 2.1

Factor $2$ out of $2 x^{2} - 12 x - 80$ .

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Step 2.1.1

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

$\frac{2 (x^{2}) - 12 x - 80}{- x - 2} \cdot \frac{4 - x^{2}}{x^{2} - 4} \cdot \frac{- x - 2}{(x - 2) (x + 4)}$

Step 2.1.2

Factor $2$ out of $- 12 x$ .

$\frac{2 (x^{2}) + 2 (- 6 x) - 80}{- x - 2} \cdot \frac{4 - x^{2}}{x^{2} - 4} \cdot \frac{- x - 2}{(x - 2) (x + 4)}$

Step 2.1.3

Factor $2$ out of $- 80$ .

$\frac{2 x^{2} + 2 (- 6 x) + 2 \cdot - 40}{- x - 2} \cdot \frac{4 - x^{2}}{x^{2} - 4} \cdot \frac{- x - 2}{(x - 2) (x + 4)}$

Step 2.1.4

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

$\frac{2 (x^{2} - 6 x) + 2 \cdot - 40}{- x - 2} \cdot \frac{4 - x^{2}}{x^{2} - 4} \cdot \frac{- x - 2}{(x - 2) (x + 4)}$

Step 2.1.5

Factor $2$ out of $2 (x^{2} - 6 x) + 2 \cdot - 40$ .

$\frac{2 (x^{2} - 6 x - 40)}{- x - 2} \cdot \frac{4 - x^{2}}{x^{2} - 4} \cdot \frac{- x - 2}{(x - 2) (x + 4)}$

Step 2.2

Factor $x^{2} - 6 x - 40$ using the AC method.

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Step 2.2.1

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 $- 40$ and whose sum is $- 6$ .

$- 10, 4$

Step 2.2.2

Write the factored form using these integers.

$\frac{2 ((x - 10) (x + 4))}{- x - 2} \cdot \frac{4 - x^{2}}{x^{2} - 4} \cdot \frac{- x - 2}{(x - 2) (x + 4)}$

$\frac{2 (x - 10) (x + 4)}{- x - 2} \cdot \frac{4 - x^{2}}{x^{2} - 4} \cdot \frac{- x - 2}{(x - 2) (x + 4)}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite $4$ as $2^{2}$ .

$\frac{2 (x - 10) (x + 4)}{- x - 2} \cdot \frac{2^{2} - x^{2}}{x^{2} - 4} \cdot \frac{- x - 2}{(x - 2) (x + 4)}$

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 $a = 2$ and $b = x$ .

$\frac{2 (x - 10) (x + 4)}{- x - 2} \cdot \frac{(2 + x) (2 - x)}{x^{2} - 4} \cdot \frac{- x - 2}{(x - 2) (x + 4)}$

Step 4

Simplify the denominator.

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Step 4.1

Rewrite $4$ as $2^{2}$ .

$\frac{2 (x - 10) (x + 4)}{- x - 2} \cdot \frac{(2 + x) (2 - x)}{x^{2} - 2^{2}} \cdot \frac{- x - 2}{(x - 2) (x + 4)}$

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 $b = 2$ .

$\frac{2 (x - 10) (x + 4)}{- x - 2} \cdot \frac{(2 + x) (2 - x)}{(x + 2) (x - 2)} \cdot \frac{- x - 2}{(x - 2) (x + 4)}$

Step 5

Reduce the expression by cancelling the common factors.

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Step 5.1

Cancel the common factor of $2 + x$ and $x + 2$ .

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Step 5.1.1

Reorder terms.

$\frac{2 (x - 10) (x + 4)}{- x - 2} \cdot \frac{(x + 2) (2 - x)}{(x + 2) (x - 2)} \cdot \frac{- x - 2}{(x - 2) (x + 4)}$

Step 5.1.2

Cancel the common factor.

$\frac{2 (x - 10) (x + 4)}{- x - 2} \cdot \frac{(x + 2) (2 - x)}{(x + 2) (x - 2)} \cdot \frac{- x - 2}{(x - 2) (x + 4)}$

Step 5.1.3

Rewrite the expression.

$\frac{2 (x - 10) (x + 4)}{- x - 2} \cdot \frac{2 - x}{x - 2} \cdot \frac{- x - 2}{(x - 2) (x + 4)}$

Step 5.2

Cancel the common factor of $2 - x$ and $x - 2$ .

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Step 5.2.1

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{2 (x - 10) (x + 4)}{- x - 2} \cdot \frac{- 1 (- 2) - x}{x - 2} \cdot \frac{- x - 2}{(x - 2) (x + 4)}$

Step 5.2.2

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

$\frac{2 (x - 10) (x + 4)}{- x - 2} \cdot \frac{- 1 (- 2) - (x)}{x - 2} \cdot \frac{- x - 2}{(x - 2) (x + 4)}$

Step 5.2.3

Factor $- 1$ out of $- 1 (- 2) - (x)$ .

$\frac{2 (x - 10) (x + 4)}{- x - 2} \cdot \frac{- 1 (- 2 + x)}{x - 2} \cdot \frac{- x - 2}{(x - 2) (x + 4)}$

Step 5.2.4

Reorder terms.

$\frac{2 (x - 10) (x + 4)}{- x - 2} \cdot \frac{- 1 (x - 2)}{x - 2} \cdot \frac{- x - 2}{(x - 2) (x + 4)}$

Step 5.2.5

Cancel the common factor.

$\frac{2 (x - 10) (x + 4)}{- x - 2} \cdot \frac{- 1 (x - 2)}{x - 2} \cdot \frac{- x - 2}{(x - 2) (x + 4)}$

Step 5.2.6

Divide $- 1$ by $1$ .

$\frac{2 (x - 10) (x + 4)}{- x - 2} \cdot - 1 \cdot \frac{- x - 2}{(x - 2) (x + 4)}$

Step 6

Multiply $\frac{2 (x - 10) (x + 4)}{- x - 2} \cdot - 1$ .

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Step 6.1

Combine $\frac{2 (x - 10) (x + 4)}{- x - 2}$ and $- 1$ .

$\frac{2 (x - 10) (x + 4) \cdot - 1}{- x - 2} \cdot \frac{- x - 2}{(x - 2) (x + 4)}$

Step 6.2

Multiply $- 1$ by $2$ .

$\frac{- 2 (x - 10) (x + 4)}{- x - 2} \cdot \frac{- x - 2}{(x - 2) (x + 4)}$

Step 7

Cancel the common factor of $x + 4$ .

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Step 7.1

Factor $x + 4$ out of $- 2 (x - 10) (x + 4)$ .

$\frac{(x + 4) (- 2 (x - 10))}{- x - 2} \cdot \frac{- x - 2}{(x - 2) (x + 4)}$

Step 7.2

Factor $x + 4$ out of $(x - 2) (x + 4)$ .

$\frac{(x + 4) (- 2 (x - 10))}{- x - 2} \cdot \frac{- x - 2}{(x + 4) (x - 2)}$

Step 7.3

Cancel the common factor.

$\frac{(x + 4) (- 2 (x - 10))}{- x - 2} \cdot \frac{- x - 2}{(x + 4) (x - 2)}$

Step 7.4

Rewrite the expression.

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

Step 8

Multiply $\frac{- 2 (x - 10)}{- x - 2}$ by $\frac{- x - 2}{x - 2}$ .

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

Step 9

Cancel the common factor of $- x - 2$ .

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Step 9.1

Cancel the common factor.

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

Step 9.2

Rewrite the expression.

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

Step 10

Move the negative in front of the fraction.

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