Algebra Examples

$\frac{\frac{5}{x} - \frac{x}{5}}{\frac{x}{25} + \frac{1}{5}}$

Step 1

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

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Step 1.1

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

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

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 $x \cdot 25$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Multiply $25$ by $5$ .

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

Step 3.3

Cancel the common factor of $5$ .

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

Cancel the common factor.

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

Step 3.3.4

Rewrite the expression.

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

Add $1$ and $1$ .

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

Step 3.8

Multiply $- 1$ by $5$ .

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

Step 3.9

Cancel the common factor of $25$ .

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Step 3.9.1

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

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

Step 3.9.2

Cancel the common factor.

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

Step 3.9.3

Rewrite the expression.

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

Step 3.10

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

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

Step 3.11

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

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

Step 3.12

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

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

Step 3.13

Add $1$ and $1$ .

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

Step 3.14

Cancel the common factor of $5$ .

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Step 3.14.1

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

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

Step 3.14.2

Cancel the common factor.

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

Step 3.14.3

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

Factor $5$ out of $125 - 5 x^{2}$ .

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Step 4.1.1

Factor $5$ out of $125$ .

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

Step 4.1.2

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

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

Step 4.1.3

Factor $5$ out of $5 (25) + 5 (- x^{2})$ .

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

Step 4.2

Rewrite $25$ as $5^{2}$ .

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

Step 4.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5$ and $b = x$ .

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

Step 5

Simplify terms.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.1.3

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

$\frac{5 (5 + x) (5 - x)}{x (x + 5)}$

Step 5.2

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

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Step 5.2.1

Reorder terms.

$\frac{5 (x + 5) (5 - x)}{x (x + 5)}$

Step 5.2.2

Cancel the common factor.

$\frac{5 (x + 5) (5 - x)}{x (x + 5)}$

Step 5.2.3

Rewrite the expression.

$\frac{5 (5 - x)}{x}$