Algebra Examples

$\frac{x^{3} - x}{2 x^{2} + 6 x} \div \frac{5 x^{2} - 5 x}{2 x + 6}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{3} - x}{2 x^{2} + 6 x} \cdot \frac{2 x + 6}{5 x^{2} - 5 x}$

Step 2

Simplify the numerator.

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Step 2.1

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

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Step 2.1.1

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

$\frac{x \cdot x^{2} - x}{2 x^{2} + 6 x} \cdot \frac{2 x + 6}{5 x^{2} - 5 x}$

Step 2.1.2

Factor $x$ out of $- x$ .

$\frac{x \cdot x^{2} + x \cdot - 1}{2 x^{2} + 6 x} \cdot \frac{2 x + 6}{5 x^{2} - 5 x}$

Step 2.1.3

Factor $x$ out of $x \cdot x^{2} + x \cdot - 1$ .

$\frac{x (x^{2} - 1)}{2 x^{2} + 6 x} \cdot \frac{2 x + 6}{5 x^{2} - 5 x}$

Step 2.2

Rewrite $1$ as $1^{2}$ .

$\frac{x (x^{2} - 1^{2})}{2 x^{2} + 6 x} \cdot \frac{2 x + 6}{5 x^{2} - 5 x}$

Step 2.3

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 $b = 1$ .

$\frac{x (x + 1) (x - 1)}{2 x^{2} + 6 x} \cdot \frac{2 x + 6}{5 x^{2} - 5 x}$

Step 3

Simplify terms.

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Step 3.1

Factor $2 x$ out of $2 x^{2} + 6 x$ .

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Step 3.1.1

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

$\frac{x (x + 1) (x - 1)}{2 x (x) + 6 x} \cdot \frac{2 x + 6}{5 x^{2} - 5 x}$

Step 3.1.2

Factor $2 x$ out of $6 x$ .

$\frac{x (x + 1) (x - 1)}{2 x (x) + 2 x (3)} \cdot \frac{2 x + 6}{5 x^{2} - 5 x}$

Step 3.1.3

Factor $2 x$ out of $2 x (x) + 2 x (3)$ .

$\frac{x (x + 1) (x - 1)}{2 x (x + 3)} \cdot \frac{2 x + 6}{5 x^{2} - 5 x}$

Step 3.2

Cancel the common factor of $x$ .

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Step 3.2.1

Cancel the common factor.

$\frac{x (x + 1) (x - 1)}{2 x (x + 3)} \cdot \frac{2 x + 6}{5 x^{2} - 5 x}$

Step 3.2.2

Rewrite the expression.

$\frac{(x + 1) (x - 1)}{2 (x + 3)} \cdot \frac{2 x + 6}{5 x^{2} - 5 x}$

Step 3.3

Factor $2$ out of $2 x + 6$ .

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Step 3.3.1

Factor $2$ out of $2 x$ .

$\frac{(x + 1) (x - 1)}{2 (x + 3)} \cdot \frac{2 (x) + 6}{5 x^{2} - 5 x}$

Step 3.3.2

Factor $2$ out of $6$ .

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

Step 3.3.3

Factor $2$ out of $2 x + 2 \cdot 3$ .

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

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

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

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

Step 3.5

Cancel the common factor of $x - 1$ .

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Step 3.5.1

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

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

Step 3.5.2

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

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

Step 3.5.3

Cancel the common factor.

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

Step 3.5.4

Rewrite the expression.

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

Step 3.6

Cancel the common factor of $2 (x + 3)$ .

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Step 3.6.1

Cancel the common factor.

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

Step 3.6.2

Rewrite the expression.

$(x + 1) \frac{1}{5 x}$

Step 3.7

Multiply $x + 1$ by $\frac{1}{5 x}$ .

$\frac{x + 1}{5 x}$