Simplify (x^2-6x-16)/x*(2x)/(x^2-4)

$\frac{x^{2} - 6 x - 16}{x} \cdot \frac{2 x}{x^{2} - 4}$

Step 1

Factor $x^{2} - 6 x - 16$ 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 $- 16$ and whose sum is $- 6$ .

$- 8, 2$

Write the factored form using these integers.

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

Step 2

Simplify the denominator.

Rewrite $4$ as $2^{2}$ .

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

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 $b = 2$ .

$\frac{(x - 8) (x + 2)}{x} \cdot \frac{2 x}{(x + 2) (x - 2)}$

Step 3

Simplify terms.

Cancel the common factor of $x + 2$ .

Factor $x + 2$ out of $(x - 8) (x + 2)$ .

$\frac{(x + 2) (x - 8)}{x} \cdot \frac{2 x}{(x + 2) (x - 2)}$

Cancel the common factor.

$\frac{(x + 2) (x - 8)}{x} \cdot \frac{2 x}{(x + 2) (x - 2)}$

Rewrite the expression.

$\frac{x - 8}{x} \cdot \frac{2 x}{x - 2}$

Cancel the common factor of $x$ .

Factor $x$ out of $2 x$ .

$\frac{x - 8}{x} \cdot \frac{x \cdot 2}{x - 2}$

Cancel the common factor.

$\frac{x - 8}{x} \cdot \frac{x \cdot 2}{x - 2}$

Rewrite the expression.

$(x - 8) \cdot \frac{2}{x - 2}$

Multiply $x - 8$ by $\frac{2}{x - 2}$ .

$\frac{(x - 8) \cdot 2}{x - 2}$

Move $2$ to the left of $x - 8$ .

$\frac{2 (x - 8)}{x - 2}$

Simplify (x^2-6x-16)/x*(2x)/(x^2-4)

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