Simplify (1/(x^2)-1/9)/(5+15/x)

$\frac{\frac{1}{x^{2}} - \frac{1}{9}}{5 + \frac{15}{x}}$

Step 1

Multiply the numerator and denominator of the complex fraction by $9 x^{2}$ .

Multiply $\frac{\frac{1}{x^{2}} - \frac{1}{9}}{5 + \frac{15}{x}}$ by $\frac{9 x^{2}}{9 x^{2}}$ .

$\frac{9 x^{2}}{9 x^{2}} \cdot \frac{\frac{1}{x^{2}} - \frac{1}{9}}{5 + \frac{15}{x}}$

Combine.

$\frac{9 x^{2} (\frac{1}{x^{2}} - \frac{1}{9})}{9 x^{2} (5 + \frac{15}{x})}$

Step 2

Apply the distributive property.

$\frac{9 x^{2} \frac{1}{x^{2}} + 9 x^{2} (- \frac{1}{9})}{9 x^{2} \cdot 5 + 9 x^{2} \frac{15}{x}}$

Step 3

Simplify by cancelling.

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

Factor $x^{2}$ out of $9 x^{2}$ .

$\frac{x^{2} \cdot 9 \frac{1}{x^{2}} + 9 x^{2} (- \frac{1}{9})}{9 x^{2} \cdot 5 + 9 x^{2} \frac{15}{x}}$

Cancel the common factor.

$\frac{x^{2} \cdot 9 \frac{1}{x^{2}} + 9 x^{2} (- \frac{1}{9})}{9 x^{2} \cdot 5 + 9 x^{2} \frac{15}{x}}$

Rewrite the expression.

$\frac{9 + 9 x^{2} (- \frac{1}{9})}{9 x^{2} \cdot 5 + 9 x^{2} \frac{15}{x}}$

Cancel the common factor of $9$ .

Move the leading negative in $- \frac{1}{9}$ into the numerator.

$\frac{9 + 9 x^{2} \frac{- 1}{9}}{9 x^{2} \cdot 5 + 9 x^{2} \frac{15}{x}}$

Factor $9$ out of $9 x^{2}$ .

$\frac{9 + 9 (x^{2}) \frac{- 1}{9}}{9 x^{2} \cdot 5 + 9 x^{2} \frac{15}{x}}$

Cancel the common factor.

$\frac{9 + 9 x^{2} \frac{- 1}{9}}{9 x^{2} \cdot 5 + 9 x^{2} \frac{15}{x}}$

Rewrite the expression.

$\frac{9 + x^{2} \cdot - 1}{9 x^{2} \cdot 5 + 9 x^{2} \frac{15}{x}}$

Cancel the common factor of $x$ .

Factor $x$ out of $9 x^{2}$ .

$\frac{9 + x^{2} \cdot - 1}{9 x^{2} \cdot 5 + x (9 x) \frac{15}{x}}$

Cancel the common factor.

$\frac{9 + x^{2} \cdot - 1}{9 x^{2} \cdot 5 + x (9 x) \frac{15}{x}}$

Rewrite the expression.

$\frac{9 + x^{2} \cdot - 1}{9 x^{2} \cdot 5 + 9 x \cdot 15}$

Multiply $15$ by $9$ .

$\frac{9 + x^{2} \cdot - 1}{9 x^{2} \cdot 5 + 135 x}$

Step 4

Simplify the numerator.

Move $- 1$ to the left of $x^{2}$ .

$\frac{9 - 1 \cdot x^{2}}{9 x^{2} \cdot 5 + 135 x}$

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$\frac{9 - x^{2}}{9 x^{2} \cdot 5 + 135 x}$

Rewrite $9 - x^{2}$ in a factored form.

Rewrite $9$ as $3^{2}$ .

$\frac{3^{2} - x^{2}}{9 x^{2} \cdot 5 + 135 x}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3$ and $b = x$ .

$\frac{(3 + x) (3 - x)}{9 x^{2} \cdot 5 + 135 x}$

Step 5

Simplify the denominator.

Factor $9 x \cdot 5$ out of $9 x^{2} \cdot 5 + 135 x$ .

Factor $9 x \cdot 5$ out of $9 x^{2} \cdot 5$ .

$\frac{(3 + x) (3 - x)}{9 x \cdot 5 (x) + 135 x}$

Factor $9 x \cdot 5$ out of $135 x$ .

$\frac{(3 + x) (3 - x)}{9 x \cdot 5 x + 9 x \cdot 5 \cdot 3}$

Factor $9 x \cdot 5$ out of $9 x \cdot 5 x + 9 x \cdot 5 \cdot 3$ .

$\frac{(3 + x) (3 - x)}{9 x \cdot 5 (x + 3)}$

Multiply $5$ by $9$ .

$\frac{(3 + x) (3 - x)}{45 x (x + 3)}$

Step 6

Cancel the common factor of $3 + x$ and $x + 3$ .

Reorder terms.

$\frac{(x + 3) (3 - x)}{45 x (x + 3)}$

Cancel the common factor.

$\frac{(x + 3) (3 - x)}{45 x (x + 3)}$

Rewrite the expression.

$\frac{3 - x}{45 x}$

Simplify (1/(x^2)-1/9)/(5+15/x)

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