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

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

Step 1

Multiply the numerator and denominator of the complex fraction by $25 x^{2}$ .

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

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

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

Cancel the common factor of $25$ .

Factor $25$ out of $25 x^{2}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x^{2}$ .

Move the leading negative in $- \frac{1}{x^{2}}$ into the numerator.

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

Factor $x^{2}$ out of $25 x^{2}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $25$ by $- 1$ .

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

Cancel the common factor of $5$ .

Factor $5$ out of $25 x^{2}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x$ .

Factor $x$ out of $25 x^{2}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

Simplify the numerator.

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

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

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 $b = 5$ .

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

Step 5

Simplify terms.

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

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

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

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

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

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

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

Cancel the common factor of $x + 5$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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