Algebra Examples

$\frac{1 - \frac{5}{x}}{\frac{x}{5} - \frac{x^{2}}{25}}$

Step 1

Multiply the numerator and denominator of the fraction by $x \cdot 25$ .

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Step 1.1

Multiply $\frac{1 - \frac{5}{x}}{\frac{x}{5} - \frac{x^{2}}{25}}$ by $\frac{x \cdot 25}{x \cdot 25}$ .

$\frac{x \cdot 25}{x \cdot 25} \cdot \frac{1 - \frac{5}{x}}{\frac{x}{5} - \frac{x^{2}}{25}}$

Step 1.2

Combine.

$\frac{x \cdot 25 (1 - \frac{5}{x})}{x \cdot 25 (\frac{x}{5} - \frac{x^{2}}{25})}$

Step 2

Apply the distributive property.

$\frac{x \cdot 25 \cdot 1 + x \cdot 25 (- \frac{5}{x})}{x \cdot 25 \frac{x}{5} + x \cdot 25 (- \frac{x^{2}}{25})}$

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

Move the leading negative in $- \frac{5}{x}$ into the numerator.

$\frac{x \cdot 25 \cdot 1 + x \cdot 25 \frac{- 5}{x}}{x \cdot 25 \frac{x}{5} + x \cdot 25 (- \frac{x^{2}}{25})}$

Step 3.1.2

Factor $x$ out of $x \cdot 25$ .

$\frac{x \cdot 25 \cdot 1 + x (25) \frac{- 5}{x}}{x \cdot 25 \frac{x}{5} + x \cdot 25 (- \frac{x^{2}}{25})}$

Step 3.1.3

Cancel the common factor.

$\frac{x \cdot 25 \cdot 1 + x \cdot 25 \frac{- 5}{x}}{x \cdot 25 \frac{x}{5} + x \cdot 25 (- \frac{x^{2}}{25})}$

Step 3.1.4

Rewrite the expression.

$\frac{x \cdot 25 \cdot 1 + 25 \cdot - 5}{x \cdot 25 \frac{x}{5} + x \cdot 25 (- \frac{x^{2}}{25})}$

Step 3.2

Multiply $25$ by $- 5$ .

$\frac{x \cdot 25 \cdot 1 - 125}{x \cdot 25 \frac{x}{5} + x \cdot 25 (- \frac{x^{2}}{25})}$

Step 3.3

Cancel the common factor of $5$ .

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Step 3.3.1

Factor $5$ out of $x \cdot 25$ .

$\frac{x \cdot 25 \cdot 1 - 125}{5 (x \cdot 5) \frac{x}{5} + x \cdot 25 (- \frac{x^{2}}{25})}$

Step 3.3.2

Cancel the common factor.

$\frac{x \cdot 25 \cdot 1 - 125}{5 (x \cdot 5) \frac{x}{5} + x \cdot 25 (- \frac{x^{2}}{25})}$

Step 3.3.3

Rewrite the expression.

$\frac{x \cdot 25 \cdot 1 - 125}{x \cdot 5 x + x \cdot 25 (- \frac{x^{2}}{25})}$

Step 3.4

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

$\frac{x \cdot 25 \cdot 1 - 125}{5 (x^{1} x) + x \cdot 25 (- \frac{x^{2}}{25})}$

Step 3.5

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

$\frac{x \cdot 25 \cdot 1 - 125}{5 (x^{1} x^{1}) + x \cdot 25 (- \frac{x^{2}}{25})}$

Step 3.6

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

$\frac{x \cdot 25 \cdot 1 - 125}{5 x^{1 + 1} + x \cdot 25 (- \frac{x^{2}}{25})}$

Step 3.7

Add $1$ and $1$ .

$\frac{x \cdot 25 \cdot 1 - 125}{5 x^{2} + x \cdot 25 (- \frac{x^{2}}{25})}$

Step 3.8

Cancel the common factor of $25$ .

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Step 3.8.1

Move the leading negative in $- \frac{x^{2}}{25}$ into the numerator.

$\frac{x \cdot 25 \cdot 1 - 125}{5 x^{2} + x \cdot 25 \frac{- x^{2}}{25}}$

Step 3.8.2

Factor $25$ out of $x \cdot 25$ .

$\frac{x \cdot 25 \cdot 1 - 125}{5 x^{2} + 25 \cdot x \frac{- x^{2}}{25}}$

Step 3.8.3

Cancel the common factor.

$\frac{x \cdot 25 \cdot 1 - 125}{5 x^{2} + 25 \cdot x \frac{- x^{2}}{25}}$

Step 3.8.4

Rewrite the expression.

$\frac{x \cdot 25 \cdot 1 - 125}{5 x^{2} + x (- x^{2})}$

Step 3.9

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

$\frac{x \cdot 25 \cdot 1 - 125}{5 x^{2} - (x^{1} x^{2})}$

Step 3.10

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

$\frac{x \cdot 25 \cdot 1 - 125}{5 x^{2} - x^{1 + 2}}$

Step 3.11

Add $1$ and $2$ .

$\frac{x \cdot 25 \cdot 1 - 125}{5 x^{2} - x^{3}}$

Step 4

Simplify the numerator.

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Step 4.1

Factor $25$ out of $x \cdot 25 \cdot 1 - 125$ .

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Step 4.1.1

Factor $25$ out of $x \cdot 25 \cdot 1$ .

$\frac{25 (x \cdot 1) - 125}{5 x^{2} - x^{3}}$

Step 4.1.2

Factor $25$ out of $- 125$ .

$\frac{25 (x \cdot 1) + 25 \cdot - 5}{5 x^{2} - x^{3}}$

Step 4.1.3

Factor $25$ out of $25 (x \cdot 1) + 25 \cdot - 5$ .

$\frac{25 (x \cdot 1 - 5)}{5 x^{2} - x^{3}}$

Step 4.2

Multiply $x$ by $1$ .

$\frac{25 (x - 5)}{5 x^{2} - x^{3}}$

Step 5

Simplify terms.

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Step 5.1

Factor $x^{2}$ out of $5 x^{2} - x^{3}$ .

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Step 5.1.1

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

$\frac{25 (x - 5)}{x^{2} \cdot 5 - x^{3}}$

Step 5.1.2

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

$\frac{25 (x - 5)}{x^{2} \cdot 5 + x^{2} (- x)}$

Step 5.1.3

Factor $x^{2}$ out of $x^{2} \cdot 5 + x^{2} (- x)$ .

$\frac{25 (x - 5)}{x^{2} (5 - x)}$

Step 5.2

Cancel the common factor of $x - 5$ and $5 - x$ .

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Step 5.2.1

Factor $- 1$ out of $x$ .

$\frac{25 (- 1 (- x) - 5)}{x^{2} (5 - x)}$

Step 5.2.2

Rewrite $- 5$ as $- 1 (5)$ .

$\frac{25 (- 1 (- x) - 1 (5))}{x^{2} (5 - x)}$

Step 5.2.3

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

$\frac{25 (- 1 (- x + 5))}{x^{2} (5 - x)}$

Step 5.2.4

Reorder terms.

$\frac{25 (- 1 (- x + 5))}{x^{2} (- x + 5)}$

Step 5.2.5

Cancel the common factor.

$\frac{25 (- 1 (- x + 5))}{x^{2} (- x + 5)}$

Step 5.2.6

Rewrite the expression.

$\frac{25 \cdot (- 1)}{x^{2}}$

Step 6

Multiply $25$ by $- 1$ .

$\frac{- 25}{x^{2}}$

Step 7

Move the negative in front of the fraction.

$- \frac{25}{x^{2}}$