Simplify (6x^2-384)/(4x^2-40x+64)*(x^2-4)/(x+8)

$\frac{6 x^{2} - 384}{4 x^{2} - 40 x + 64} \cdot \frac{x^{2} - 4}{x + 8}$

Step 1

Simplify the numerator.

Factor $6$ out of $6 x^{2} - 384$ .

Factor $6$ out of $6 x^{2}$ .

$\frac{6 (x^{2}) - 384}{4 x^{2} - 40 x + 64} \cdot \frac{x^{2} - 4}{x + 8}$

Factor $6$ out of $- 384$ .

$\frac{6 x^{2} + 6 \cdot - 64}{4 x^{2} - 40 x + 64} \cdot \frac{x^{2} - 4}{x + 8}$

Factor $6$ out of $6 x^{2} + 6 \cdot - 64$ .

$\frac{6 (x^{2} - 64)}{4 x^{2} - 40 x + 64} \cdot \frac{x^{2} - 4}{x + 8}$

Rewrite $64$ as $8^{2}$ .

$\frac{6 (x^{2} - 8^{2})}{4 x^{2} - 40 x + 64} \cdot \frac{x^{2} - 4}{x + 8}$

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 $b = 8$ .

$\frac{6 (x + 8) (x - 8)}{4 x^{2} - 40 x + 64} \cdot \frac{x^{2} - 4}{x + 8}$

Step 2

Simplify the denominator.

Factor $4$ out of $4 x^{2} - 40 x + 64$ .

Factor $4$ out of $4 x^{2}$ .

$\frac{6 (x + 8) (x - 8)}{4 (x^{2}) - 40 x + 64} \cdot \frac{x^{2} - 4}{x + 8}$

Factor $4$ out of $- 40 x$ .

$\frac{6 (x + 8) (x - 8)}{4 (x^{2}) + 4 (- 10 x) + 64} \cdot \frac{x^{2} - 4}{x + 8}$

Factor $4$ out of $64$ .

$\frac{6 (x + 8) (x - 8)}{4 x^{2} + 4 (- 10 x) + 4 \cdot 16} \cdot \frac{x^{2} - 4}{x + 8}$

Factor $4$ out of $4 x^{2} + 4 (- 10 x)$ .

$\frac{6 (x + 8) (x - 8)}{4 (x^{2} - 10 x) + 4 \cdot 16} \cdot \frac{x^{2} - 4}{x + 8}$

Factor $4$ out of $4 (x^{2} - 10 x) + 4 \cdot 16$ .

$\frac{6 (x + 8) (x - 8)}{4 (x^{2} - 10 x + 16)} \cdot \frac{x^{2} - 4}{x + 8}$

Factor $x^{2} - 10 x + 16$ 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 $- 10$ .

$- 8, - 2$

Write the factored form using these integers.

$\frac{6 (x + 8) (x - 8)}{4 ((x - 8) (x - 2))} \cdot \frac{x^{2} - 4}{x + 8}$

$\frac{6 (x + 8) (x - 8)}{4 (x - 8) (x - 2)} \cdot \frac{x^{2} - 4}{x + 8}$

Step 3

Simplify the numerator.

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

$\frac{6 (x + 8) (x - 8)}{4 (x - 8) (x - 2)} \cdot \frac{x^{2} - 2^{2}}{x + 8}$

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 $b = 2$ .

$\frac{6 (x + 8) (x - 8)}{4 (x - 8) (x - 2)} \cdot \frac{(x + 2) (x - 2)}{x + 8}$

Step 4

Simplify terms.

Cancel the common factor of $x + 8$ .

Factor $x + 8$ out of $6 (x + 8) (x - 8)$ .

$\frac{(x + 8) (6 (x - 8))}{4 (x - 8) (x - 2)} \cdot \frac{(x + 2) (x - 2)}{x + 8}$

Cancel the common factor.

$\frac{(x + 8) (6 (x - 8))}{4 (x - 8) (x - 2)} \cdot \frac{(x + 2) (x - 2)}{x + 8}$

Rewrite the expression.

$\frac{6 (x - 8)}{4 (x - 8) (x - 2)} \cdot ((x + 2) (x - 2))$

Cancel the common factor of $x - 2$ .

Factor $x - 2$ out of $4 (x - 8) (x - 2)$ .

$\frac{6 (x - 8)}{(x - 2) (4 (x - 8))} \cdot ((x + 2) (x - 2))$

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

$\frac{6 (x - 8)}{(x - 2) (4 (x - 8))} \cdot ((x - 2) (x + 2))$

Cancel the common factor.

$\frac{6 (x - 8)}{(x - 2) (4 (x - 8))} \cdot ((x - 2) (x + 2))$

Rewrite the expression.

$\frac{6 (x - 8)}{4 (x - 8)} \cdot (x + 2)$

Cancel the common factor of $6$ and $4$ .

Factor $2$ out of $6 (x - 8)$ .

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

Cancel the common factors.

Factor $2$ out of $4 (x - 8)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x - 8$ .

Cancel the common factor.

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

Rewrite the expression.

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

Apply the distributive property.

$\frac{3}{2} x + \frac{3}{2} \cdot 2$

Combine $\frac{3}{2}$ and $x$ .

$\frac{3 x}{2} + \frac{3}{2} \cdot 2$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{3 x}{2} + \frac{3}{2} \cdot 2$

Rewrite the expression.

$\frac{3 x}{2} + 3$

Step 5

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 x}{2} + 3 \cdot \frac{2}{2}$

Step 6

Combine $3$ and $\frac{2}{2}$ .

$\frac{3 x}{2} + \frac{3 \cdot 2}{2}$

Step 7

Combine the numerators over the common denominator.

$\frac{3 x + 3 \cdot 2}{2}$

Step 8

Factor $3$ out of $3 x + 3 \cdot 2$ .

Factor $3$ out of $3 x$ .

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

Factor $3$ out of $3 \cdot 2$ .

$\frac{3 (x) + 3 (2)}{2}$

Factor $3$ out of $3 (x) + 3 (2)$ .

$\frac{3 (x + 2)}{2}$

Simplify (6x^2-384)/(4x^2-40x+64)*(x^2-4)/(x+8)

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