Simplify (2x)/(x-4)+64/(x^2-16)-(2x)/(x+4)

$\frac{2 x}{x - 4} + \frac{64}{x^{2} - 16} - \frac{2 x}{x + 4}$

Step 1

Simplify the denominator.

Rewrite $16$ as $4^{2}$ .

$\frac{2 x}{x - 4} + \frac{64}{x^{2} - 4^{2}} - \frac{2 x}{x + 4}$

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 $b = 4$ .

$\frac{2 x}{x - 4} + \frac{64}{(x + 4) (x - 4)} - \frac{2 x}{x + 4}$

Step 2

To write $\frac{2 x}{x - 4}$ as a fraction with a common denominator, multiply by $\frac{x + 4}{x + 4}$ .

$\frac{2 x}{x - 4} \cdot \frac{x + 4}{x + 4} + \frac{64}{(x + 4) (x - 4)} - \frac{2 x}{x + 4}$

Step 3

Write each expression with a common denominator of $(x - 4) (x + 4)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{2 x}{x - 4}$ by $\frac{x + 4}{x + 4}$ .

$\frac{2 x (x + 4)}{(x - 4) (x + 4)} + \frac{64}{(x + 4) (x - 4)} - \frac{2 x}{x + 4}$

Reorder the factors of $(x - 4) (x + 4)$ .

$\frac{2 x (x + 4)}{(x + 4) (x - 4)} + \frac{64}{(x + 4) (x - 4)} - \frac{2 x}{x + 4}$

Step 4

Combine the numerators over the common denominator.

$\frac{2 x (x + 4) + 64}{(x + 4) (x - 4)} - \frac{2 x}{x + 4}$

Step 5

Simplify the numerator.

Factor $2$ out of $2 x (x + 4) + 64$ .

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

$\frac{2 (x (x + 4)) + 64}{(x + 4) (x - 4)} - \frac{2 x}{x + 4}$

Factor $2$ out of $64$ .

$\frac{2 (x (x + 4)) + 2 \cdot 32}{(x + 4) (x - 4)} - \frac{2 x}{x + 4}$

Factor $2$ out of $2 (x (x + 4)) + 2 \cdot 32$ .

$\frac{2 (x (x + 4) + 32)}{(x + 4) (x - 4)} - \frac{2 x}{x + 4}$

Apply the distributive property.

$\frac{2 (x \cdot x + x \cdot 4 + 32)}{(x + 4) (x - 4)} - \frac{2 x}{x + 4}$

Multiply $x$ by $x$ .

$\frac{2 (x^{2} + x \cdot 4 + 32)}{(x + 4) (x - 4)} - \frac{2 x}{x + 4}$

Move $4$ to the left of $x$ .

$\frac{2 (x^{2} + 4 x + 32)}{(x + 4) (x - 4)} - \frac{2 x}{x + 4}$

Step 6

To write $- \frac{2 x}{x + 4}$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

$\frac{2 (x^{2} + 4 x + 32)}{(x + 4) (x - 4)} - \frac{2 x}{x + 4} \cdot \frac{x - 4}{x - 4}$

Step 7

Simplify terms.

Multiply $\frac{2 x}{x + 4}$ by $\frac{x - 4}{x - 4}$ .

$\frac{2 (x^{2} + 4 x + 32)}{(x + 4) (x - 4)} - \frac{2 x (x - 4)}{(x + 4) (x - 4)}$

Combine the numerators over the common denominator.

$\frac{2 (x^{2} + 4 x + 32) - 2 x (x - 4)}{(x + 4) (x - 4)}$

Step 8

Simplify the numerator.

Factor $2$ out of $2 (x^{2} + 4 x + 32) - 2 x (x - 4)$ .

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

$\frac{2 (x^{2} + 4 x + 32) + 2 (- x (x - 4))}{(x + 4) (x - 4)}$

Factor $2$ out of $2 (x^{2} + 4 x + 32) + 2 (- x (x - 4))$ .

$\frac{2 (x^{2} + 4 x + 32 - x (x - 4))}{(x + 4) (x - 4)}$

Apply the distributive property.

$\frac{2 (x^{2} + 4 x + 32 - x \cdot x - x \cdot - 4)}{(x + 4) (x - 4)}$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$\frac{2 (x^{2} + 4 x + 32 - (x \cdot x) - x \cdot - 4)}{(x + 4) (x - 4)}$

Multiply $x$ by $x$ .

$\frac{2 (x^{2} + 4 x + 32 - x^{2} - x \cdot - 4)}{(x + 4) (x - 4)}$

Multiply $- 4$ by $- 1$ .

$\frac{2 (x^{2} + 4 x + 32 - x^{2} + 4 x)}{(x + 4) (x - 4)}$

Subtract $x^{2}$ from $x^{2}$ .

$\frac{2 (4 x + 32 + 0 + 4 x)}{(x + 4) (x - 4)}$

Add $4 x$ and $0$ .

$\frac{2 (4 x + 32 + 4 x)}{(x + 4) (x - 4)}$

Add $4 x$ and $4 x$ .

$\frac{2 (8 x + 32)}{(x + 4) (x - 4)}$

Factor $8$ out of $8 x + 32$ .

Factor $8$ out of $8 x$ .

$\frac{2 (8 (x) + 32)}{(x + 4) (x - 4)}$

Factor $8$ out of $32$ .

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

Factor $8$ out of $8 x + 8 \cdot 4$ .

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

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

Multiply $2$ by $8$ .

$\frac{16 (x + 4)}{(x + 4) (x - 4)}$

Step 9

Cancel the common factor of $x + 4$ .

Cancel the common factor.

$\frac{16 (x + 4)}{(x + 4) (x - 4)}$

Rewrite the expression.

$\frac{16}{x - 4}$

Simplify (2x)/(x-4)+64/(x^2-16)-(2x)/(x+4)

Related Questions

Support LCMGCF.com by disabling your adblocker.