Algebra Examples

$\frac{x^{2} - 9}{x^{2} - 5 x} \cdot \frac{5 x - x^{2}}{x^{2} - x - 12} \div \frac{x - 4}{x^{2} - 8 x + 16}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 9}{x^{2} - 5 x} \cdot \frac{5 x - x^{2}}{x^{2} - x - 12} \frac{x^{2} - 8 x + 16}{x - 4}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $9$ as $3^{2}$ .

$\frac{x^{2} - 3^{2}}{x^{2} - 5 x} \cdot \frac{5 x - x^{2}}{x^{2} - x - 12} \frac{x^{2} - 8 x + 16}{x - 4}$

Step 2.2

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 $b = 3$ .

$\frac{(x + 3) (x - 3)}{x^{2} - 5 x} \cdot \frac{5 x - x^{2}}{x^{2} - x - 12} \frac{x^{2} - 8 x + 16}{x - 4}$

Step 3

Simplify with factoring out.

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Step 3.1

Factor $x$ out of $x^{2} - 5 x$ .

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Step 3.1.1

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

$\frac{(x + 3) (x - 3)}{x \cdot x - 5 x} \cdot \frac{5 x - x^{2}}{x^{2} - x - 12} \frac{x^{2} - 8 x + 16}{x - 4}$

Step 3.1.2

Factor $x$ out of $- 5 x$ .

$\frac{(x + 3) (x - 3)}{x \cdot x + x \cdot - 5} \cdot \frac{5 x - x^{2}}{x^{2} - x - 12} \frac{x^{2} - 8 x + 16}{x - 4}$

Step 3.1.3

Factor $x$ out of $x \cdot x + x \cdot - 5$ .

$\frac{(x + 3) (x - 3)}{x (x - 5)} \cdot \frac{5 x - x^{2}}{x^{2} - x - 12} \frac{x^{2} - 8 x + 16}{x - 4}$

Step 3.2

Factor $x$ out of $5 x - x^{2}$ .

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Step 3.2.1

Factor $x$ out of $5 x$ .

$\frac{(x + 3) (x - 3)}{x (x - 5)} \cdot \frac{x \cdot 5 - x^{2}}{x^{2} - x - 12} \frac{x^{2} - 8 x + 16}{x - 4}$

Step 3.2.2

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

$\frac{(x + 3) (x - 3)}{x (x - 5)} \cdot \frac{x \cdot 5 + x (- x)}{x^{2} - x - 12} \frac{x^{2} - 8 x + 16}{x - 4}$

Step 3.2.3

Factor $x$ out of $x \cdot 5 + x (- x)$ .

$\frac{(x + 3) (x - 3)}{x (x - 5)} \cdot \frac{x (5 - x)}{x^{2} - x - 12} \frac{x^{2} - 8 x + 16}{x - 4}$

Step 4

Factor $x^{2} - x - 12$ using the AC method.

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Step 4.1

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 $- 1$ .

$- 4, 3$

Step 4.2

Write the factored form using these integers.

$\frac{(x + 3) (x - 3)}{x (x - 5)} \cdot \frac{x (5 - x)}{(x - 4) (x + 3)} \frac{x^{2} - 8 x + 16}{x - 4}$

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $x + 3$ .

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Step 5.1.1

Factor $x + 3$ out of $(x - 4) (x + 3)$ .

$\frac{(x + 3) (x - 3)}{x (x - 5)} \cdot \frac{x (5 - x)}{(x + 3) (x - 4)} \frac{x^{2} - 8 x + 16}{x - 4}$

Step 5.1.2

Cancel the common factor.

$\frac{(x + 3) (x - 3)}{x (x - 5)} \cdot \frac{x (5 - x)}{(x + 3) (x - 4)} \frac{x^{2} - 8 x + 16}{x - 4}$

Step 5.1.3

Rewrite the expression.

$\frac{x - 3}{x (x - 5)} \cdot \frac{x (5 - x)}{x - 4} \frac{x^{2} - 8 x + 16}{x - 4}$

Step 5.2

Cancel the common factor of $x$ .

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Step 5.2.1

Cancel the common factor.

$\frac{x - 3}{x (x - 5)} \cdot \frac{x (5 - x)}{x - 4} \frac{x^{2} - 8 x + 16}{x - 4}$

Step 5.2.2

Rewrite the expression.

$\frac{x - 3}{x - 5} \cdot \frac{5 - x}{x - 4} \frac{x^{2} - 8 x + 16}{x - 4}$

Step 5.3

Multiply $\frac{x - 3}{x - 5}$ by $\frac{5 - x}{x - 4}$ .

$\frac{(x - 3) (5 - x)}{(x - 5) (x - 4)} \cdot \frac{x^{2} - 8 x + 16}{x - 4}$

Step 5.4

Cancel the common factor of $5 - x$ and $x - 5$ .

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Step 5.4.1

Rewrite $5$ as $- 1 (- 5)$ .

$\frac{(x - 3) (- 1 (- 5) - x)}{(x - 5) (x - 4)} \cdot \frac{x^{2} - 8 x + 16}{x - 4}$

Step 5.4.2

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

$\frac{(x - 3) (- 1 (- 5) - (x))}{(x - 5) (x - 4)} \cdot \frac{x^{2} - 8 x + 16}{x - 4}$

Step 5.4.3

Factor $- 1$ out of $- 1 (- 5) - (x)$ .

$\frac{(x - 3) (- 1 (- 5 + x))}{(x - 5) (x - 4)} \cdot \frac{x^{2} - 8 x + 16}{x - 4}$

Step 5.4.4

Reorder terms.

$\frac{(x - 3) (- 1 (x - 5))}{(x - 5) (x - 4)} \cdot \frac{x^{2} - 8 x + 16}{x - 4}$

Step 5.4.5

Cancel the common factor.

$\frac{(x - 3) (- 1 (x - 5))}{(x - 5) (x - 4)} \cdot \frac{x^{2} - 8 x + 16}{x - 4}$

Step 5.4.6

Rewrite the expression.

$\frac{(x - 3) \cdot (- 1)}{x - 4} \cdot \frac{x^{2} - 8 x + 16}{x - 4}$

Step 5.5

Simplify the expression.

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Step 5.5.1

Move $- 1$ to the left of $x - 3$ .

$\frac{- 1 \cdot (x - 3)}{x - 4} \cdot \frac{x^{2} - 8 x + 16}{x - 4}$

Step 5.5.2

Move the negative in front of the fraction.

$- \frac{x - 3}{x - 4} \cdot \frac{x^{2} - 8 x + 16}{x - 4}$

Step 6

Factor using the perfect square rule.

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Step 6.1

Rewrite $16$ as $4^{2}$ .

$- \frac{x - 3}{x - 4} \cdot \frac{x^{2} - 8 x + 4^{2}}{x - 4}$

Step 6.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$8 x = 2 \cdot x \cdot 4$

Step 6.3

Rewrite the polynomial.

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

Step 6.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 4$ .

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

Step 7

Simplify terms.

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Step 7.1

Cancel the common factor of $x - 4$ .

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Step 7.1.1

Move the leading negative in $- \frac{x - 3}{x - 4}$ into the numerator.

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

Step 7.1.2

Factor $x - 4$ out of ${(x - 4)}^{2}$ .

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

Step 7.1.3

Cancel the common factor.

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

Step 7.1.4

Rewrite the expression.

$- (x - 3) \frac{x - 4}{x - 4}$

Step 7.2

Cancel the common factor of $x - 4$ .

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Step 7.2.1

Cancel the common factor.

$- (x - 3) \frac{x - 4}{x - 4}$

Step 7.2.2

Rewrite the expression.

$- (x - 3) \cdot 1$

Step 7.3

Multiply $- (x - 3)$ by $1$ .

$- (x - 3)$

Step 7.4

Apply the distributive property.

$- x - - 3$

Step 7.5

Multiply $- 1$ by $- 3$ .

$- x + 3$