Simplify ((3x^2-9x)/(x-2))÷((x^2-9)/(4x-8))

$\frac{3 x^{2} - 9 x}{x - 2} \div \frac{x^{2} - 9}{4 x - 8}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{3 x^{2} - 9 x}{x - 2} \cdot \frac{4 x - 8}{x^{2} - 9}$

Step 2

Simplify with factoring out.

Factor $3 x$ out of $3 x^{2} - 9 x$ .

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

$\frac{3 x (x) - 9 x}{x - 2} \cdot \frac{4 x - 8}{x^{2} - 9}$

Factor $3 x$ out of $- 9 x$ .

$\frac{3 x (x) + 3 x (- 3)}{x - 2} \cdot \frac{4 x - 8}{x^{2} - 9}$

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

$\frac{3 x (x - 3)}{x - 2} \cdot \frac{4 x - 8}{x^{2} - 9}$

Factor $4$ out of $4 x - 8$ .

Factor $4$ out of $4 x$ .

$\frac{3 x (x - 3)}{x - 2} \cdot \frac{4 (x) - 8}{x^{2} - 9}$

Factor $4$ out of $- 8$ .

$\frac{3 x (x - 3)}{x - 2} \cdot \frac{4 x + 4 \cdot - 2}{x^{2} - 9}$

Factor $4$ out of $4 x + 4 \cdot - 2$ .

$\frac{3 x (x - 3)}{x - 2} \cdot \frac{4 (x - 2)}{x^{2} - 9}$

Step 3

Simplify the denominator.

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

$\frac{3 x (x - 3)}{x - 2} \cdot \frac{4 (x - 2)}{x^{2} - 3^{2}}$

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 $b = 3$ .

$\frac{3 x (x - 3)}{x - 2} \cdot \frac{4 (x - 2)}{(x + 3) (x - 3)}$

Step 4

Simplify terms.

Cancel the common factor of $x - 3$ .

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

$\frac{(x - 3) (3 x)}{x - 2} \cdot \frac{4 (x - 2)}{(x + 3) (x - 3)}$

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

$\frac{(x - 3) (3 x)}{x - 2} \cdot \frac{4 (x - 2)}{(x - 3) (x + 3)}$

Cancel the common factor.

$\frac{(x - 3) (3 x)}{x - 2} \cdot \frac{4 (x - 2)}{(x - 3) (x + 3)}$

Rewrite the expression.

$\frac{3 x}{x - 2} \cdot \frac{4 (x - 2)}{x + 3}$

Cancel the common factor of $x - 2$ .

Factor $x - 2$ out of $4 (x - 2)$ .

$\frac{3 x}{x - 2} \cdot \frac{(x - 2) \cdot 4}{x + 3}$

Cancel the common factor.

$\frac{3 x}{x - 2} \cdot \frac{(x - 2) \cdot 4}{x + 3}$

Rewrite the expression.

$3 x \frac{4}{x + 3}$

Combine $3$ and $\frac{4}{x + 3}$ .

$x \frac{3 \cdot 4}{x + 3}$

Multiply $3$ by $4$ .

$x \frac{12}{x + 3}$

Combine $x$ and $\frac{12}{x + 3}$ .

$\frac{x \cdot 12}{x + 3}$

Move $12$ to the left of $x$ .

$\frac{12 x}{x + 3}$

Simplify ((3x^2-9x)/(x-2))÷((x^2-9)/(4x-8))

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