Algebra Examples

$\frac{x^{2} + 4 x - 45}{- x - 4} \cdot \frac{x + 5}{x^{2} + 14 x + 45} \cdot \frac{- 9}{x^{2} - 5 x}$

Step 1

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

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Step 1.1

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

$\frac{x^{2} + 4 x - 45}{- x - 4} \cdot \frac{x + 5}{x^{2} + 14 x + 45} \cdot \frac{- 9}{x \cdot x - 5 x}$

Step 1.2

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

$\frac{x^{2} + 4 x - 45}{- x - 4} \cdot \frac{x + 5}{x^{2} + 14 x + 45} \cdot \frac{- 9}{x \cdot x + x \cdot - 5}$

Step 1.3

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

$\frac{x^{2} + 4 x - 45}{- x - 4} \cdot \frac{x + 5}{x^{2} + 14 x + 45} \cdot \frac{- 9}{x (x - 5)}$

Step 2

Factor $x^{2} + 4 x - 45$ 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 $- 45$ and whose sum is $4$ .

$- 5, 9$

Step 2.2

Write the factored form using these integers.

$\frac{(x - 5) (x + 9)}{- x - 4} \cdot \frac{x + 5}{x^{2} + 14 x + 45} \cdot \frac{- 9}{x (x - 5)}$

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 $14$ .

$5, 9$

Step 3.2

Write the factored form using these integers.

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

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $x + 9$ .

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

Cancel the common factor.

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

Step 4.1.4

Rewrite the expression.

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

Step 4.2

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

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

Step 4.3

Cancel the common factor of $x - 5$ .

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Step 4.3.1

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

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

Step 4.3.2

Cancel the common factor.

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

Step 4.3.3

Rewrite the expression.

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

Step 4.4

Multiply $\frac{x + 5}{(- x - 4) (x + 5)}$ by $\frac{- 9}{x}$ .

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

Step 4.5

Cancel the common factor of $x + 5$ .

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Step 4.5.1

Cancel the common factor.

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

Step 4.5.2

Rewrite the expression.

$\frac{- 9}{(- x - 4) x}$

Step 4.6

Move the negative in front of the fraction.

$- \frac{9}{(- x - 4) x}$

Step 4.7

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

$- \frac{9}{(- (x) - 4) x}$

Step 4.8

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 4.9

Factor $- 1$ out of $- (x) - 1 (4)$ .

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

Step 4.10

Simplify the expression.

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Step 4.10.1

Rewrite $- (x + 4)$ as $- 1 (x + 4)$ .

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

Step 4.10.2

Move the negative in front of the fraction.

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

Step 4.10.3

Multiply $- 1$ by $- 1$ .

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

Step 4.10.4

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

$\frac{9}{(x + 4) x}$

Step 4.10.5

Reorder factors in $\frac{9}{(x + 4) x}$ .

$\frac{9}{x (x + 4)}$