Simplify (2+10/x)/(1-(x^2)/25)

$\frac{2 + \frac{10}{x}}{1 - \frac{x^{2}}{25}}$

Step 1

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

Multiply $\frac{2 + \frac{10}{x}}{1 - \frac{x^{2}}{25}}$ by $\frac{x \cdot 25}{x \cdot 25}$ .

$\frac{x \cdot 25}{x \cdot 25} \cdot \frac{2 + \frac{10}{x}}{1 - \frac{x^{2}}{25}}$

Combine.

$\frac{x \cdot 25 (2 + \frac{10}{x})}{x \cdot 25 (1 - \frac{x^{2}}{25})}$

Step 2

Apply the distributive property.

$\frac{x \cdot 25 \cdot 2 + x \cdot 25 \frac{10}{x}}{x \cdot 25 \cdot 1 + x \cdot 25 (- \frac{x^{2}}{25})}$

Step 3

Simplify by cancelling.

Cancel the common factor of $x$ .

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

$\frac{x \cdot 25 \cdot 2 + x (25) \frac{10}{x}}{x \cdot 25 \cdot 1 + x \cdot 25 (- \frac{x^{2}}{25})}$

Cancel the common factor.

$\frac{x \cdot 25 \cdot 2 + x \cdot 25 \frac{10}{x}}{x \cdot 25 \cdot 1 + x \cdot 25 (- \frac{x^{2}}{25})}$

Rewrite the expression.

$\frac{x \cdot 25 \cdot 2 + 25 \cdot 10}{x \cdot 25 \cdot 1 + x \cdot 25 (- \frac{x^{2}}{25})}$

Multiply $25$ by $10$ .

$\frac{x \cdot 25 \cdot 2 + 250}{x \cdot 25 \cdot 1 + x \cdot 25 (- \frac{x^{2}}{25})}$

Cancel the common factor of $25$ .

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

$\frac{x \cdot 25 \cdot 2 + 250}{x \cdot 25 \cdot 1 + x \cdot 25 \frac{- x^{2}}{25}}$

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

$\frac{x \cdot 25 \cdot 2 + 250}{x \cdot 25 \cdot 1 + 25 \cdot x \frac{- x^{2}}{25}}$

Cancel the common factor.

$\frac{x \cdot 25 \cdot 2 + 250}{x \cdot 25 \cdot 1 + 25 \cdot x \frac{- x^{2}}{25}}$

Rewrite the expression.

$\frac{x \cdot 25 \cdot 2 + 250}{x \cdot 25 \cdot 1 + x (- x^{2})}$

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

$\frac{x \cdot 25 \cdot 2 + 250}{x \cdot 25 \cdot 1 - (x^{1} x^{2})}$

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

$\frac{x \cdot 25 \cdot 2 + 250}{x \cdot 25 \cdot 1 - x^{1 + 2}}$

Add $1$ and $2$ .

$\frac{x \cdot 25 \cdot 2 + 250}{x \cdot 25 \cdot 1 - x^{3}}$

Step 4

Simplify the numerator.

Move $25$ to the left of $x$ .

$\frac{25 \cdot x \cdot 2 + 250}{x \cdot 25 \cdot 1 - x^{3}}$

Multiply $2$ by $25$ .

$\frac{50 x + 250}{x \cdot 25 \cdot 1 - x^{3}}$

Factor $50$ out of $50 x + 250$ .

Factor $50$ out of $50 x$ .

$\frac{50 (x) + 250}{x \cdot 25 \cdot 1 - x^{3}}$

Factor $50$ out of $250$ .

$\frac{50 x + 50 \cdot 5}{x \cdot 25 \cdot 1 - x^{3}}$

Factor $50$ out of $50 x + 50 \cdot 5$ .

$\frac{50 (x + 5)}{x \cdot 25 \cdot 1 - x^{3}}$

Step 5

Simplify the denominator.

Factor $x$ out of $x \cdot 25 \cdot 1 - x^{3}$ .

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

$\frac{50 (x + 5)}{x (25 \cdot 1) - x^{3}}$

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

$\frac{50 (x + 5)}{x (25 \cdot 1) + x (- x^{2})}$

Factor $x$ out of $x (25 \cdot 1) + x (- x^{2})$ .

$\frac{50 (x + 5)}{x (25 \cdot 1 - x^{2})}$

Rewrite $25 \cdot 1$ as ${(5 \cdot 1)}^{2}$ .

$\frac{50 (x + 5)}{x ({(5 \cdot 1)}^{2} - x^{2})}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5 \cdot 1$ and $b = x$ .

$\frac{50 (x + 5)}{x ((5 \cdot 1 + x) (5 \cdot 1 - x))}$

Simplify.

Multiply $5$ by $1$ .

$\frac{50 (x + 5)}{x ((5 + x) (5 \cdot 1 - x))}$

Multiply $5$ by $1$ .

$\frac{50 (x + 5)}{x ((5 + x) (5 - x))}$

$\frac{50 (x + 5)}{x (5 + x) (5 - x)}$

Step 6

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

Reorder terms.

$\frac{50 (x + 5)}{x (x + 5) (5 - x)}$

Cancel the common factor.

$\frac{50 (x + 5)}{x (x + 5) (5 - x)}$

Rewrite the expression.

$\frac{50}{x (5 - x)}$

Simplify (2+10/x)/(1-(x^2)/25)

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