Simplify (9/x-3)/(3-(x^2)/3)

$\frac{\frac{9}{x} - 3}{3 - \frac{x^{2}}{3}}$

Step 1

Multiply the numerator and denominator of the complex fraction by $x \cdot 3$ .

Multiply $\frac{\frac{9}{x} - 3}{3 - \frac{x^{2}}{3}}$ by $\frac{x \cdot 3}{x \cdot 3}$ .

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

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

Cancel the common factor of $x$ .

Factor $x$ out of $x \cdot 3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $3$ by $9$ .

$\frac{27 + x \cdot 3 \cdot - 3}{x \cdot 3 \cdot 3 + x \cdot 3 (- \frac{x^{2}}{3})}$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{x^{2}}{3}$ into the numerator.

$\frac{27 + x \cdot 3 \cdot - 3}{x \cdot 3 \cdot 3 + x \cdot 3 \frac{- x^{2}}{3}}$

Factor $3$ out of $x \cdot 3$ .

$\frac{27 + x \cdot 3 \cdot - 3}{x \cdot 3 \cdot 3 + 3 \cdot x \frac{- x^{2}}{3}}$

Cancel the common factor.

$\frac{27 + x \cdot 3 \cdot - 3}{x \cdot 3 \cdot 3 + 3 \cdot x \frac{- x^{2}}{3}}$

Rewrite the expression.

$\frac{27 + x \cdot 3 \cdot - 3}{x \cdot 3 \cdot 3 + x (- x^{2})}$

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

$\frac{27 + x \cdot 3 \cdot - 3}{x \cdot 3 \cdot 3 - (x^{1} x^{2})}$

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

$\frac{27 + x \cdot 3 \cdot - 3}{x \cdot 3 \cdot 3 - x^{1 + 2}}$

Add $1$ and $2$ .

$\frac{27 + x \cdot 3 \cdot - 3}{x \cdot 3 \cdot 3 - x^{3}}$

Step 4

Simplify the numerator.

Move $3$ to the left of $x$ .

$\frac{27 + 3 \cdot x \cdot - 3}{x \cdot 3 \cdot 3 - x^{3}}$

Multiply $- 3$ by $3$ .

$\frac{27 - 9 x}{x \cdot 3 \cdot 3 - x^{3}}$

Factor $9$ out of $27 - 9 x$ .

Factor $9$ out of $27$ .

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

Factor $9$ out of $- 9 x$ .

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

Factor $9$ out of $9 (3) + 9 (- x)$ .

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

Step 5

Simplify the denominator.

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

Factor $x$ out of $x \cdot 3 \cdot 3$ .

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

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

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

Factor $x$ out of $x (3 \cdot 3) + x (- x^{2})$ .

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

Multiply $3$ by $3$ .

$\frac{9 (3 - x)}{x (9 - x^{2})}$

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

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

$\frac{9 (3 - x)}{x (3^{2} - x^{2})}$

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

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

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

Step 6

Cancel the common factor of $3 - x$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{9}{x (3 + x)}$

Simplify (9/x-3)/(3-(x^2)/3)

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