Algebra Examples

$\frac{\frac{1}{2} - \frac{x}{4}}{\frac{x^{2}}{4} - 1}$

Step 1

Multiply the numerator and denominator of the fraction by $4$ .

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Step 1.1

Multiply $\frac{\frac{1}{2} - \frac{x}{4}}{\frac{x^{2}}{4} - 1}$ by $\frac{4}{4}$ .

$\frac{4}{4} \cdot \frac{\frac{1}{2} - \frac{x}{4}}{\frac{x^{2}}{4} - 1}$

Step 1.2

Combine.

$\frac{4 (\frac{1}{2} - \frac{x}{4})}{4 (\frac{x^{2}}{4} - 1)}$

Step 2

Apply the distributive property.

$\frac{4 (\frac{1}{2}) + 4 (- \frac{x}{4})}{4 \frac{x^{2}}{4} + 4 \cdot - 1}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $2$ .

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Step 3.1.1

Factor $2$ out of $4$ .

$\frac{2 (2) \frac{1}{2} + 4 (- \frac{x}{4})}{4 \frac{x^{2}}{4} + 4 \cdot - 1}$

Step 3.1.2

Cancel the common factor.

$\frac{2 \cdot 2 \frac{1}{2} + 4 (- \frac{x}{4})}{4 \frac{x^{2}}{4} + 4 \cdot - 1}$

Step 3.1.3

Rewrite the expression.

$\frac{2 + 4 (- \frac{x}{4})}{4 \frac{x^{2}}{4} + 4 \cdot - 1}$

Step 3.2

Cancel the common factor of $4$ .

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Step 3.2.1

Move the leading negative in $- \frac{x}{4}$ into the numerator.

$\frac{2 + 4 \frac{- x}{4}}{4 \frac{x^{2}}{4} + 4 \cdot - 1}$

Step 3.2.2

Cancel the common factor.

$\frac{2 + 4 \frac{- x}{4}}{4 \frac{x^{2}}{4} + 4 \cdot - 1}$

Step 3.2.3

Rewrite the expression.

$\frac{2 - x}{4 \frac{x^{2}}{4} + 4 \cdot - 1}$

Step 3.3

Cancel the common factor of $4$ .

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Step 3.3.1

Cancel the common factor.

$\frac{2 - x}{4 \frac{x^{2}}{4} + 4 \cdot - 1}$

Step 3.3.2

Rewrite the expression.

$\frac{2 - x}{x^{2} + 4 \cdot - 1}$

Step 4

Simplify the denominator.

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Step 4.1

Multiply $4$ by $- 1$ .

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

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 $b = 2$ .

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

Step 5

Reduce the expression by cancelling the common factors.

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Step 5.1

Cancel the common factor of $2 - x$ and $x - 2$ .

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Step 5.1.1

Rewrite $2$ as $- 1 (- 2)$ .

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

Step 5.1.2

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

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

Step 5.1.3

Factor $- 1$ out of $- 1 (- 2) - (x)$ .

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

Step 5.1.4

Reorder terms.

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

Step 5.1.5

Cancel the common factor.

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

Step 5.1.6

Rewrite the expression.

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

Step 5.2

Move the negative in front of the fraction.

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