Algebra Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

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 $4 x$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $x$ .

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Step 3.2.1

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

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

Step 3.2.2

Factor $x$ out of $4 x$ .

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

Step 3.2.3

Cancel the common factor.

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

Step 3.2.4

Rewrite the expression.

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

Step 3.3

Multiply $4$ by $- 1$ .

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

Step 3.4

Cancel the common factor of $4$ .

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Step 3.4.1

Factor $4$ out of $4 x$ .

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

Step 3.4.2

Cancel the common factor.

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

Step 3.4.3

Rewrite the expression.

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

Step 3.5

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

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Step 3.5.1

Multiply $x$ by $x^{2}$ .

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Step 3.5.1.1

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

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

Step 3.5.1.2

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

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

Step 3.5.2

Add $1$ and $2$ .

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

Step 3.6

Cancel the common factor of $2$ .

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Step 3.6.1

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

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

Step 3.6.2

Factor $2$ out of $4 x$ .

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

Step 3.6.3

Cancel the common factor.

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

Step 3.6.4

Rewrite the expression.

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

Step 3.7

Multiply $- 1$ by $2$ .

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

Step 3.8

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

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

Step 3.9

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

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

Step 3.10

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

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

Step 3.11

Add $1$ and $1$ .

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

Step 4

Simplify the numerator.

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Step 4.1

Factor $2$ out of $2 x (x - 1) - 4$ .

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Step 4.1.1

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

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

Step 4.1.2

Factor $2$ out of $- 4$ .

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

Step 4.1.3

Factor $2$ out of $2 (x (x - 1)) + 2 \cdot - 2$ .

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Multiply $x$ by $x$ .

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

Step 4.4

Move $- 1$ to the left of $x$ .

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

Step 4.5

Rewrite $- 1 x$ as $- x$ .

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

Step 4.6

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

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Step 4.6.1

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 $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Step 4.6.2

Write the factored form using these integers.

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

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

Step 5

Simplify terms.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.1.3

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

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

Step 5.2

Cancel the common factor of $x - 2$ .

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Step 5.2.1

Cancel the common factor.

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

Step 5.2.2

Rewrite the expression.

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