Algebra Examples

$\frac{2}{x + 3} \cdot \frac{- x - 5}{x^{3} + 2 x^{2} - 15 x} \div \frac{- 2}{9 - x^{2}}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{2}{x + 3} \cdot \frac{- x - 5}{x^{3} + 2 x^{2} - 15 x} \frac{9 - x^{2}}{- 2}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $x$ out of $x^{3} + 2 x^{2} - 15 x$ .

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Step 2.1.1

Factor $x$ out of $x^{3}$ .

$\frac{2}{x + 3} \cdot \frac{- x - 5}{x \cdot x^{2} + 2 x^{2} - 15 x} \frac{9 - x^{2}}{- 2}$

Step 2.1.2

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

$\frac{2}{x + 3} \cdot \frac{- x - 5}{x \cdot x^{2} + x (2 x) - 15 x} \frac{9 - x^{2}}{- 2}$

Step 2.1.3

Factor $x$ out of $- 15 x$ .

$\frac{2}{x + 3} \cdot \frac{- x - 5}{x \cdot x^{2} + x (2 x) + x \cdot - 15} \frac{9 - x^{2}}{- 2}$

Step 2.1.4

Factor $x$ out of $x \cdot x^{2} + x (2 x)$ .

$\frac{2}{x + 3} \cdot \frac{- x - 5}{x (x^{2} + 2 x) + x \cdot - 15} \frac{9 - x^{2}}{- 2}$

Step 2.1.5

Factor $x$ out of $x (x^{2} + 2 x) + x \cdot - 15$ .

$\frac{2}{x + 3} \cdot \frac{- x - 5}{x (x^{2} + 2 x - 15)} \frac{9 - x^{2}}{- 2}$

Step 2.2

Factor $x^{2} + 2 x - 15$ using the AC method.

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Step 2.2.1

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 $- 15$ and whose sum is $2$ .

$- 3, 5$

Step 2.2.2

Write the factored form using these integers.

$\frac{2}{x + 3} \cdot \frac{- x - 5}{x ((x - 3) (x + 5))} \frac{9 - x^{2}}{- 2}$

$\frac{2}{x + 3} \cdot \frac{- x - 5}{x (x - 3) (x + 5)} \frac{9 - x^{2}}{- 2}$

Step 3

Simplify terms.

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Step 3.1

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

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Step 3.1.1

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

$\frac{2}{x + 3} \cdot \frac{- (x) - 5}{x (x - 3) (x + 5)} \frac{9 - x^{2}}{- 2}$

Step 3.1.2

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

$\frac{2}{x + 3} \cdot \frac{- (x) - 1 (5)}{x (x - 3) (x + 5)} \frac{9 - x^{2}}{- 2}$

Step 3.1.3

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

$\frac{2}{x + 3} \cdot \frac{- (x + 5)}{x (x - 3) (x + 5)} \frac{9 - x^{2}}{- 2}$

Step 3.1.4

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

$\frac{2}{x + 3} \cdot \frac{- 1 (x + 5)}{x (x - 3) (x + 5)} \frac{9 - x^{2}}{- 2}$

Step 3.1.5

Cancel the common factor.

$\frac{2}{x + 3} \cdot \frac{- 1 (x + 5)}{x (x - 3) (x + 5)} \frac{9 - x^{2}}{- 2}$

Step 3.1.6

Rewrite the expression.

$\frac{2}{x + 3} \cdot \frac{- 1}{x (x - 3)} \frac{9 - x^{2}}{- 2}$

Step 3.2

Simplify the expression.

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Step 3.2.1

Move the negative in front of the fraction.

$\frac{2}{x + 3} \cdot (- \frac{1}{x (x - 3)}) \frac{9 - x^{2}}{- 2}$

Step 3.2.2

Rewrite using the commutative property of multiplication.

$- \frac{2}{x + 3} \cdot \frac{1}{x (x - 3)} \frac{9 - x^{2}}{- 2}$

Step 3.3

Multiply $\frac{1}{x (x - 3)}$ by $\frac{2}{x + 3}$ .

$- \frac{2}{x (x - 3) (x + 3)} \cdot \frac{9 - x^{2}}{- 2}$

Step 3.4

Cancel the common factor of $2$ .

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Step 3.4.1

Move the leading negative in $- \frac{2}{x (x - 3) (x + 3)}$ into the numerator.

$\frac{- 2}{x (x - 3) (x + 3)} \cdot \frac{9 - x^{2}}{- 2}$

Step 3.4.2

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{x (x - 3) (x + 3)} \cdot \frac{9 - x^{2}}{- 2}$

Step 3.4.3

Factor $2$ out of $- 2$ .

$\frac{2 \cdot - 1}{x (x - 3) (x + 3)} \cdot \frac{9 - x^{2}}{2 \cdot - 1}$

Step 3.4.4

Cancel the common factor.

$\frac{2 \cdot - 1}{x (x - 3) (x + 3)} \cdot \frac{9 - x^{2}}{2 \cdot - 1}$

Step 3.4.5

Rewrite the expression.

$\frac{- 1}{x (x - 3) (x + 3)} \cdot \frac{9 - x^{2}}{- 1}$

Step 3.5

Multiply $\frac{- 1}{x (x - 3) (x + 3)}$ by $\frac{9 - x^{2}}{- 1}$ .

$\frac{- (9 - x^{2})}{x (x - 3) (x + 3) \cdot - 1}$

Step 3.6

Dividing two negative values results in a positive value.

$\frac{9 - x^{2}}{x (x - 3) (x + 3)}$

Step 4

Simplify the numerator.

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Step 4.1

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

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

Step 4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3$ and $b = x$ .

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

Step 5

Reduce the expression by cancelling the common factors.

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Step 5.1

Cancel the common factor of $3 + x$ and $x + 3$ .

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Step 5.1.1

Reorder terms.

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

Step 5.1.2

Cancel the common factor.

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

Step 5.1.3

Rewrite the expression.

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

Step 5.2

Cancel the common factor of $3 - x$ and $x - 3$ .

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Step 5.2.1

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{- 1 (- 3) - x}{x (x - 3)}$

Step 5.2.2

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

$\frac{- 1 (- 3) - (x)}{x (x - 3)}$

Step 5.2.3

Factor $- 1$ out of $- 1 (- 3) - (x)$ .

$\frac{- 1 (- 3 + x)}{x (x - 3)}$

Step 5.2.4

Reorder terms.

$\frac{- 1 (x - 3)}{x (x - 3)}$

Step 5.2.5

Cancel the common factor.

$\frac{- 1 (x - 3)}{x (x - 3)}$

Step 5.2.6

Rewrite the expression.

$\frac{- 1}{x}$

Step 5.3

Move the negative in front of the fraction.

$- \frac{1}{x}$