Simplify (1/x-x/25)/((x^2)/25+x/5)

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

Step 1

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

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

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

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

Cancel the common factor of $x$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $25$ .

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Cancel the common factor of $25$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

Add $1$ and $2$ .

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

Cancel the common factor of $5$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Step 4

Simplify the numerator.

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

$\frac{5^{2} - x^{2}}{x^{3} + 5 x^{2}}$

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 $b = x$ .

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

Step 5

Simplify terms.

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

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

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

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

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

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

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

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

Reorder terms.

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify (1/x-x/25)/((x^2)/25+x/5)

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