Algebra Examples

$\frac{x^{2} - 81}{x^{2} - 14 x + 45} \cdot \frac{x^{2} - 12 x + 35}{- x - 9} \div \frac{7 - x}{- 4}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 81}{x^{2} - 14 x + 45} \cdot \frac{x^{2} - 12 x + 35}{- x - 9} \frac{- 4}{7 - x}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $81$ as $9^{2}$ .

$\frac{x^{2} - 9^{2}}{x^{2} - 14 x + 45} \cdot \frac{x^{2} - 12 x + 35}{- x - 9} \frac{- 4}{7 - x}$

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 $b = 9$ .

$\frac{(x + 9) (x - 9)}{x^{2} - 14 x + 45} \cdot \frac{x^{2} - 12 x + 35}{- x - 9} \frac{- 4}{7 - x}$

Step 3

Factor $x^{2} - 14 x + 45$ 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 $45$ and whose sum is $- 14$ .

$- 9, - 5$

Step 3.2

Write the factored form using these integers.

$\frac{(x + 9) (x - 9)}{(x - 9) (x - 5)} \cdot \frac{x^{2} - 12 x + 35}{- x - 9} \frac{- 4}{7 - x}$

Step 4

Factor $x^{2} - 12 x + 35$ 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 $35$ and whose sum is $- 12$ .

$- 7, - 5$

Step 4.2

Write the factored form using these integers.

$\frac{(x + 9) (x - 9)}{(x - 9) (x - 5)} \cdot \frac{(x - 7) (x - 5)}{- x - 9} \frac{- 4}{7 - x}$

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $x - 5$ .

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Step 5.1.1

Factor $x - 5$ out of $(x - 9) (x - 5)$ .

$\frac{(x + 9) (x - 9)}{(x - 5) (x - 9)} \cdot \frac{(x - 7) (x - 5)}{- x - 9} \frac{- 4}{7 - x}$

Step 5.1.2

Factor $x - 5$ out of $(x - 7) (x - 5)$ .

$\frac{(x + 9) (x - 9)}{(x - 5) (x - 9)} \cdot \frac{(x - 5) (x - 7)}{- x - 9} \frac{- 4}{7 - x}$

Step 5.1.3

Cancel the common factor.

$\frac{(x + 9) (x - 9)}{(x - 5) (x - 9)} \cdot \frac{(x - 5) (x - 7)}{- x - 9} \frac{- 4}{7 - x}$

Step 5.1.4

Rewrite the expression.

$\frac{(x + 9) (x - 9)}{x - 9} \cdot \frac{x - 7}{- x - 9} \frac{- 4}{7 - x}$

Step 5.2

Multiply $\frac{(x + 9) (x - 9)}{x - 9}$ by $\frac{x - 7}{- x - 9}$ .

$\frac{(x + 9) (x - 9) (x - 7)}{(x - 9) (- x - 9)} \cdot \frac{- 4}{7 - x}$

Step 5.3

Cancel the common factor of $x - 9$ .

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Step 5.3.1

Cancel the common factor.

$\frac{(x + 9) (x - 9) (x - 7)}{(x - 9) (- x - 9)} \cdot \frac{- 4}{7 - x}$

Step 5.3.2

Rewrite the expression.

$\frac{(x + 9) (x - 7)}{- x - 9} \cdot \frac{- 4}{7 - x}$

Step 5.4

Cancel the common factor of $x + 9$ and $- x - 9$ .

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Step 5.4.1

Factor $- 1$ out of $x$ .

$\frac{(- 1 (- x) + 9) (x - 7)}{- x - 9} \cdot \frac{- 4}{7 - x}$

Step 5.4.2

Rewrite $9$ as $- 1 (- 9)$ .

$\frac{(- 1 (- x) - 1 (- 9)) (x - 7)}{- x - 9} \cdot \frac{- 4}{7 - x}$

Step 5.4.3

Factor $- 1$ out of $- 1 (- x) - 1 (- 9)$ .

$\frac{- 1 (- x - 9) (x - 7)}{- x - 9} \cdot \frac{- 4}{7 - x}$

Step 5.4.4

Cancel the common factor.

$\frac{- 1 (- x - 9) (x - 7)}{- x - 9} \cdot \frac{- 4}{7 - x}$

Step 5.4.5

Divide $- 1 (x - 7)$ by $1$ .

$- 1 (x - 7) \frac{- 4}{7 - x}$

Step 5.5

Rewrite $- 1 (x - 7)$ as $- (x - 7)$ .

$- (x - 7) \frac{- 4}{7 - x}$

Step 5.6

Apply the distributive property.

$(- x - - 7) \frac{- 4}{7 - x}$

Step 5.7

Simplify the expression.

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Step 5.7.1

Multiply $- 1$ by $- 7$ .

$(- x + 7) \frac{- 4}{7 - x}$

Step 5.7.2

Move the negative in front of the fraction.

$(- x + 7) (- \frac{4}{7 - x})$

Step 5.8

Apply the distributive property.

$- x (- \frac{4}{7 - x}) + 7 (- \frac{4}{7 - x})$

Step 6

Multiply $- x (- \frac{4}{7 - x})$ .

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Step 6.1

Multiply $- 1$ by $- 1$ .

$1 x \frac{4}{7 - x} + 7 (- \frac{4}{7 - x})$

Step 6.2

Multiply $x$ by $1$ .

$x \frac{4}{7 - x} + 7 (- \frac{4}{7 - x})$

Step 6.3

Combine $x$ and $\frac{4}{7 - x}$ .

$\frac{x \cdot 4}{7 - x} + 7 (- \frac{4}{7 - x})$

Step 7

Multiply $7 (- \frac{4}{7 - x})$ .

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Step 7.1

Multiply $- 1$ by $7$ .

$\frac{x \cdot 4}{7 - x} - 7 \frac{4}{7 - x}$

Step 7.2

Combine $- 7$ and $\frac{4}{7 - x}$ .

$\frac{x \cdot 4}{7 - x} + \frac{- 7 \cdot 4}{7 - x}$

Step 7.3

Multiply $- 7$ by $4$ .

$\frac{x \cdot 4}{7 - x} + \frac{- 28}{7 - x}$

Step 8

Combine the numerators over the common denominator.

$\frac{x \cdot 4 - 28}{7 - x}$

Step 9

Reorder factors in $x \cdot 4 - 28$ .

$\frac{4 x - 28}{7 - x}$

Step 10

Factor $4$ out of $4 x - 28$ .

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Step 10.1

Factor $4$ out of $4 x$ .

$\frac{4 (x) - 28}{7 - x}$

Step 10.2

Factor $4$ out of $- 28$ .

$\frac{4 x + 4 \cdot - 7}{7 - x}$

Step 10.3

Factor $4$ out of $4 x + 4 \cdot - 7$ .

$\frac{4 (x - 7)}{7 - x}$

Step 11

Cancel the common factor of $x - 7$ and $7 - x$ .

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Step 11.1

Factor $- 1$ out of $x$ .

$\frac{4 (- 1 (- x) - 7)}{7 - x}$

Step 11.2

Rewrite $- 7$ as $- 1 (7)$ .

$\frac{4 (- 1 (- x) - 1 (7))}{7 - x}$

Step 11.3

Factor $- 1$ out of $- 1 (- x) - 1 (7)$ .

$\frac{4 (- 1 (- x + 7))}{7 - x}$

Step 11.4

Reorder terms.

$\frac{4 (- 1 (- x + 7))}{- x + 7}$

Step 11.5

Cancel the common factor.

$\frac{4 (- 1 (- x + 7))}{- x + 7}$

Step 11.6

Divide $4 \cdot (- 1)$ by $1$ .

$4 \cdot (- 1)$

Step 12

Multiply $4$ by $- 1$ .

$- 4$