Algebra Examples

$\frac{x^{2} - 7 x + 12}{x^{2} + x - 12} \cdot \frac{- 4 x}{- x - 2} \div \frac{x - 4}{x + 4}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 7 x + 12}{x^{2} + x - 12} \cdot \frac{- 4 x}{- x - 2} \frac{x + 4}{x - 4}$

Step 2

Factor $x^{2} - 7 x + 12$ 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 $12$ and whose sum is $- 7$ .

$- 4, - 3$

Step 2.2

Write the factored form using these integers.

$\frac{(x - 4) (x - 3)}{x^{2} + x - 12} \cdot \frac{- 4 x}{- x - 2} \frac{x + 4}{x - 4}$

Step 3

Factor $x^{2} + x - 12$ 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 $- 12$ and whose sum is $1$ .

$- 3, 4$

Step 3.2

Write the factored form using these integers.

$\frac{(x - 4) (x - 3)}{(x - 3) (x + 4)} \cdot \frac{- 4 x}{- x - 2} \frac{x + 4}{x - 4}$

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $x - 3$ .

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Step 4.1.1

Cancel the common factor.

$\frac{(x - 4) (x - 3)}{(x - 3) (x + 4)} \cdot \frac{- 4 x}{- x - 2} \frac{x + 4}{x - 4}$

Step 4.1.2

Rewrite the expression.

$\frac{x - 4}{x + 4} \cdot \frac{- 4 x}{- x - 2} \frac{x + 4}{x - 4}$

Step 4.2

Simplify the expression.

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Step 4.2.1

Move the negative in front of the fraction.

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

Step 4.2.2

Rewrite using the commutative property of multiplication.

$- \frac{x - 4}{x + 4} \cdot \frac{4 x}{- x - 2} \frac{x + 4}{x - 4}$

Step 4.3

Multiply $\frac{4 x}{- x - 2}$ by $\frac{x - 4}{x + 4}$ .

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

Step 4.4

Cancel the common factor of $x - 4$ .

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Step 4.4.1

Move the leading negative in $- \frac{4 x (x - 4)}{(- x - 2) (x + 4)}$ into the numerator.

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

Step 4.4.2

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

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

Step 4.4.3

Cancel the common factor.

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

Step 4.4.4

Rewrite the expression.

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

Step 4.5

Cancel the common factor of $x + 4$ .

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Step 4.5.1

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

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

Step 4.5.2

Cancel the common factor.

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

Step 4.5.3

Rewrite the expression.

$\frac{- 4 x}{- x - 2}$

Step 4.6

Move the negative in front of the fraction.

$- \frac{4 x}{- x - 2}$

Step 4.7

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

$- \frac{4 x}{- (x) - 2}$

Step 4.8

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

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

Step 4.9

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

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

Step 4.10

Simplify the expression.

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Step 4.10.1

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

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

Step 4.10.2

Move the negative in front of the fraction.

$- - \frac{4 x}{x + 2}$

Step 4.10.3

Multiply $- 1$ by $- 1$ .

$1 \frac{4 x}{x + 2}$

Step 4.10.4

Multiply $\frac{4 x}{x + 2}$ by $1$ .

$\frac{4 x}{x + 2}$