Simplify (x^2-3x-54)/(x^2-9x)*(2x)/(x^2+8x+12)

$\frac{x^{2} - 3 x - 54}{x^{2} - 9 x} \cdot \frac{2 x}{x^{2} + 8 x + 12}$

Step 1

Factor $x^{2} - 3 x - 54$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 54$ and whose sum is $- 3$ .

$- 9, 6$

Write the factored form using these integers.

$\frac{(x - 9) (x + 6)}{x^{2} - 9 x} \cdot \frac{2 x}{x^{2} + 8 x + 12}$

Step 2

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

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

$\frac{(x - 9) (x + 6)}{x \cdot x - 9 x} \cdot \frac{2 x}{x^{2} + 8 x + 12}$

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

$\frac{(x - 9) (x + 6)}{x \cdot x + x \cdot - 9} \cdot \frac{2 x}{x^{2} + 8 x + 12}$

Factor $x$ out of $x \cdot x + x \cdot - 9$ .

$\frac{(x - 9) (x + 6)}{x (x - 9)} \cdot \frac{2 x}{x^{2} + 8 x + 12}$

Step 3

Factor $x^{2} + 8 x + 12$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $12$ and whose sum is $8$ .

$2, 6$

Write the factored form using these integers.

$\frac{(x - 9) (x + 6)}{x (x - 9)} \cdot \frac{2 x}{(x + 2) (x + 6)}$

Step 4

Simplify terms.

Cancel the common factor of $x + 6$ .

Factor $x + 6$ out of $(x - 9) (x + 6)$ .

$\frac{(x + 6) (x - 9)}{x (x - 9)} \cdot \frac{2 x}{(x + 2) (x + 6)}$

Factor $x + 6$ out of $(x + 2) (x + 6)$ .

$\frac{(x + 6) (x - 9)}{x (x - 9)} \cdot \frac{2 x}{(x + 6) (x + 2)}$

Cancel the common factor.

$\frac{(x + 6) (x - 9)}{x (x - 9)} \cdot \frac{2 x}{(x + 6) (x + 2)}$

Rewrite the expression.

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

Cancel the common factor of $x$ .

Factor $x$ out of $2 x$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{x - 9}{x - 9} \cdot \frac{2}{x + 2}$

Multiply $\frac{x - 9}{x - 9}$ by $\frac{2}{x + 2}$ .

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

Cancel the common factor of $x - 9$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{2}{x + 2}$

Simplify (x^2-3x-54)/(x^2-9x)*(2x)/(x^2+8x+12)

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