Algebra Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

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 $4 x$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Raise $x$ to the power of $1$ .

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

Step 3.3

Raise $x$ to the power of $1$ .

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

Step 3.4

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

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

Step 3.5

Add $1$ and $1$ .

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

Step 3.6

Cancel the common factor of $4$ .

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Step 3.6.1

Move the leading negative in $- \frac{x^{2}}{4}$ into the numerator.

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

Step 3.6.2

Factor $4$ out of $4 x$ .

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

Step 3.6.3

Cancel the common factor.

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

Step 3.6.4

Rewrite the expression.

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

Step 3.7

Raise $x$ to the power of $1$ .

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

Step 3.8

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

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

Step 3.9

Add $1$ and $2$ .

$\frac{2 x^{2} - x^{3}}{4 x \frac{x}{4} + 4 x (- \frac{1}{x})}$

Step 3.10

Cancel the common factor of $4$ .

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Step 3.10.1

Factor $4$ out of $4 x$ .

$\frac{2 x^{2} - x^{3}}{4 (x) \frac{x}{4} + 4 x (- \frac{1}{x})}$

Step 3.10.2

Cancel the common factor.

$\frac{2 x^{2} - x^{3}}{4 x \frac{x}{4} + 4 x (- \frac{1}{x})}$

Step 3.10.3

Rewrite the expression.

$\frac{2 x^{2} - x^{3}}{x \cdot x + 4 x (- \frac{1}{x})}$

Step 3.11

Raise $x$ to the power of $1$ .

$\frac{2 x^{2} - x^{3}}{x^{1} x + 4 x (- \frac{1}{x})}$

Step 3.12

Raise $x$ to the power of $1$ .

$\frac{2 x^{2} - x^{3}}{x^{1} x^{1} + 4 x (- \frac{1}{x})}$

Step 3.13

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

$\frac{2 x^{2} - x^{3}}{x^{1 + 1} + 4 x (- \frac{1}{x})}$

Step 3.14

Add $1$ and $1$ .

$\frac{2 x^{2} - x^{3}}{x^{2} + 4 x (- \frac{1}{x})}$

Step 3.15

Cancel the common factor of $x$ .

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Step 3.15.1

Move the leading negative in $- \frac{1}{x}$ into the numerator.

$\frac{2 x^{2} - x^{3}}{x^{2} + 4 x \frac{- 1}{x}}$

Step 3.15.2

Factor $x$ out of $4 x$ .

$\frac{2 x^{2} - x^{3}}{x^{2} + x \cdot 4 \frac{- 1}{x}}$

Step 3.15.3

Cancel the common factor.

$\frac{2 x^{2} - x^{3}}{x^{2} + x \cdot 4 \frac{- 1}{x}}$

Step 3.15.4

Rewrite the expression.

$\frac{2 x^{2} - x^{3}}{x^{2} + 4 \cdot - 1}$

Step 3.16

Multiply $4$ by $- 1$ .

$\frac{2 x^{2} - x^{3}}{x^{2} - 4}$

Step 4

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

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Step 4.1

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

$\frac{x^{2} \cdot 2 - x^{3}}{x^{2} - 4}$

Step 4.2

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

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

Step 4.3

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

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

Step 5

Simplify the denominator.

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Step 5.1

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

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

Step 5.2

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 $b = 2$ .

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

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

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

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Step 6.1.1

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

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

Step 6.1.2

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

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

Step 6.1.3

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

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

Step 6.1.4

Reorder terms.

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

Step 6.1.5

Cancel the common factor.

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

Step 6.1.6

Rewrite the expression.

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

Step 6.2

Simplify the expression.

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Step 6.2.1

Move $- 1$ to the left of $x^{2}$ .

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

Step 6.2.2

Move the negative in front of the fraction.

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