Algebra Examples

$\frac{\frac{1}{x} - \frac{2}{x^{2}}}{\frac{x^{2}}{2} - 2}$

Step 1

Multiply the numerator and denominator of the fraction by $2 x^{2}$ .

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Step 1.1

Multiply $\frac{\frac{1}{x} - \frac{2}{x^{2}}}{\frac{x^{2}}{2} - 2}$ by $\frac{2 x^{2}}{2 x^{2}}$ .

$\frac{2 x^{2}}{2 x^{2}} \cdot \frac{\frac{1}{x} - \frac{2}{x^{2}}}{\frac{x^{2}}{2} - 2}$

Step 1.2

Combine.

$\frac{2 x^{2} (\frac{1}{x} - \frac{2}{x^{2}})}{2 x^{2} (\frac{x^{2}}{2} - 2)}$

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x$ .

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Step 3.1.1

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

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $x^{2}$ .

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Step 3.2.1

Move the leading negative in $- \frac{2}{x^{2}}$ into the numerator.

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

Step 3.2.2

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

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

Step 3.2.3

Cancel the common factor.

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

Step 3.2.4

Rewrite the expression.

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

Step 3.3

Multiply $2$ by $- 2$ .

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

Step 3.4

Cancel the common factor of $2$ .

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Step 3.4.1

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

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

Step 3.4.2

Cancel the common factor.

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

Step 3.4.3

Rewrite the expression.

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

Step 3.5

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 3.5.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.5.2

Add $2$ and $2$ .

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

Step 4

Factor $2$ out of $2 x - 4$ .

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Step 4.1

Factor $2$ out of $2 x$ .

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

Step 4.2

Factor $2$ out of $- 4$ .

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

Step 4.3

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

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

Step 5

Simplify the denominator.

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Step 5.1

Factor $x^{2}$ out of $x^{4} + 2 x^{2} \cdot - 2$ .

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Step 5.1.1

Factor $x^{2}$ out of $x^{4}$ .

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

Step 5.1.2

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

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

Step 5.1.3

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

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

Step 5.2

Multiply $2$ by $- 2$ .

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

Step 5.3

Rewrite $x^{2} - 4$ in a factored form.

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Step 5.3.1

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

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

Step 5.3.2

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 $b = 2$ .

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

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

Step 6

Cancel the common factor of $x - 2$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

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