Algebra Examples

$\frac{x^{2} - 5 x - 6}{x - 9} \cdot \frac{6 x^{2} + 24 x}{- x - 1} \div \frac{x^{2} - 2 x - 24}{- 2}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 5 x - 6}{x - 9} \cdot \frac{6 x^{2} + 24 x}{- x - 1} \frac{- 2}{x^{2} - 2 x - 24}$

Step 2

Factor $x^{2} - 5 x - 6$ 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 $- 6$ and whose sum is $- 5$ .

$- 6, 1$

Step 2.2

Write the factored form using these integers.

$\frac{(x - 6) (x + 1)}{x - 9} \cdot \frac{6 x^{2} + 24 x}{- x - 1} \frac{- 2}{x^{2} - 2 x - 24}$

Step 3

Simplify terms.

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Step 3.1

Factor $6 x$ out of $6 x^{2} + 24 x$ .

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Step 3.1.1

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

$\frac{(x - 6) (x + 1)}{x - 9} \cdot \frac{6 x (x) + 24 x}{- x - 1} \frac{- 2}{x^{2} - 2 x - 24}$

Step 3.1.2

Factor $6 x$ out of $24 x$ .

$\frac{(x - 6) (x + 1)}{x - 9} \cdot \frac{6 x (x) + 6 x (4)}{- x - 1} \frac{- 2}{x^{2} - 2 x - 24}$

Step 3.1.3

Factor $6 x$ out of $6 x (x) + 6 x (4)$ .

$\frac{(x - 6) (x + 1)}{x - 9} \cdot \frac{6 x (x + 4)}{- x - 1} \frac{- 2}{x^{2} - 2 x - 24}$

Step 3.2

Multiply $\frac{(x - 6) (x + 1)}{x - 9}$ by $\frac{6 x (x + 4)}{- x - 1}$ .

$\frac{(x - 6) (x + 1) (6 x (x + 4))}{(x - 9) (- x - 1)} \cdot \frac{- 2}{x^{2} - 2 x - 24}$

Step 3.3

Cancel the common factor of $x + 1$ and $- x - 1$ .

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Step 3.3.1

Factor $- 1$ out of $x$ .

$\frac{(x - 6) (- 1 (- x) + 1) (6 x (x + 4))}{(x - 9) (- x - 1)} \cdot \frac{- 2}{x^{2} - 2 x - 24}$

Step 3.3.2

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

$\frac{(x - 6) (- 1 (- x) - 1 (- 1)) (6 x (x + 4))}{(x - 9) (- x - 1)} \cdot \frac{- 2}{x^{2} - 2 x - 24}$

Step 3.3.3

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

$\frac{(x - 6) (- 1 (- x - 1)) (6 x (x + 4))}{(x - 9) (- x - 1)} \cdot \frac{- 2}{x^{2} - 2 x - 24}$

Step 3.3.4

Cancel the common factor.

$\frac{(x - 6) (- 1 (- x - 1)) (6 x (x + 4))}{(x - 9) (- x - 1)} \cdot \frac{- 2}{x^{2} - 2 x - 24}$

Step 3.3.5

Rewrite the expression.

$\frac{((x - 6) \cdot (- 1)) (6 x (x + 4))}{x - 9} \cdot \frac{- 2}{x^{2} - 2 x - 24}$

Step 3.4

Multiply $6$ by $- 1$ .

$\frac{(x - 6) \cdot - 6 (x (x + 4))}{x - 9} \cdot \frac{- 2}{x^{2} - 2 x - 24}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite.

$\frac{(x - 6) (x + 1) (6 x (x + 4))}{x - 9} \cdot \frac{- 2}{x^{2} - 2 x - 24}$

Step 4.2

Factor $- 1$ out of $x$ .

$\frac{(x - 6) (- 1 (- x) + 1) (6 x (x + 4))}{x - 9} \cdot \frac{- 2}{x^{2} - 2 x - 24}$

Step 4.3

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

$\frac{(x - 6) (- 1 (- x) - 1 (- 1)) (6 x (x + 4))}{x - 9} \cdot \frac{- 2}{x^{2} - 2 x - 24}$

Step 4.4

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

$\frac{(x - 6) (- 1 (- x - 1)) (6 x (x + 4))}{x - 9} \cdot \frac{- 2}{x^{2} - 2 x - 24}$

Step 4.5

Rewrite.

$\frac{(x - 6) {(- 1)}^{- 1} (6 x (x + 4))}{x - 9} \cdot \frac{- 2}{x^{2} - 2 x - 24}$

Step 4.6

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

$\frac{(x - 6) \cdot - 1 (6 x (x + 4))}{x - 9} \cdot \frac{- 2}{x^{2} - 2 x - 24}$

Step 4.7

Multiply $(x - 6) \cdot (- 1)$ by $6 x (x + 4)$ .

$\frac{((x - 6) \cdot (- 1)) (6 x (x + 4))}{x - 9} \cdot \frac{- 2}{x^{2} - 2 x - 24}$

Step 4.8

Remove unnecessary parentheses.

$\frac{(x - 6) \cdot - 6 x (x + 4)}{x - 9} \cdot \frac{- 2}{x^{2} - 2 x - 24}$

Step 5

Simplify the expression.

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Step 5.1

Move $- 6$ to the left of $x - 6$ .

$\frac{- 6 \cdot (x - 6) x (x + 4)}{x - 9} \cdot \frac{- 2}{x^{2} - 2 x - 24}$

Step 5.2

Move the negative in front of the fraction.

$- \frac{(6 \cdot (x - 6)) x (x + 4)}{x - 9} \cdot \frac{- 2}{x^{2} - 2 x - 24}$

Step 6

Factor $x^{2} - 2 x - 24$ 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 $- 24$ and whose sum is $- 2$ .

$- 6, 4$

Step 6.2

Write the factored form using these integers.

$- \frac{6 (x - 6) x (x + 4)}{x - 9} \cdot \frac{- 2}{(x - 6) (x + 4)}$

Step 7

Simplify terms.

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Step 7.1

Cancel the common factor of $(x - 6) (x + 4)$ .

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Step 7.1.1

Move the leading negative in $- \frac{6 (x - 6) x (x + 4)}{x - 9}$ into the numerator.

$\frac{- 6 (x - 6) x (x + 4)}{x - 9} \cdot \frac{- 2}{(x - 6) (x + 4)}$

Step 7.1.2

Factor $(x - 6) (x + 4)$ out of $- 6 (x - 6) x (x + 4)$ .

$\frac{(x - 6) (x + 4) (- 6 x)}{x - 9} \cdot \frac{- 2}{(x - 6) (x + 4)}$

Step 7.1.3

Cancel the common factor.

$\frac{(x - 6) (x + 4) (- 6 x)}{x - 9} \cdot \frac{- 2}{(x - 6) (x + 4)}$

Step 7.1.4

Rewrite the expression.

$\frac{- 6 x}{x - 9} \cdot - 2$

Step 7.2

Combine $\frac{- 6 x}{x - 9}$ and $- 2$ .

$\frac{- 6 x \cdot - 2}{x - 9}$

Step 7.3

Multiply $- 2$ by $- 6$ .

$\frac{12 x}{x - 9}$