Simplify (2x)/(x+2)+5/(x-2)-16/(x^2-4)

$\frac{2 x}{x + 2} + \frac{5}{x - 2} - \frac{16}{x^{2} - 4}$

Step 1

Simplify the denominator.

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

$\frac{2 x}{x + 2} + \frac{5}{x - 2} - \frac{16}{x^{2} - 2^{2}}$

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 $b = 2$ .

$\frac{2 x}{x + 2} + \frac{5}{x - 2} - \frac{16}{(x + 2) (x - 2)}$

Step 2

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

$\frac{2 x}{x + 2} \cdot \frac{x - 2}{x - 2} + \frac{5}{x - 2} - \frac{16}{(x + 2) (x - 2)}$

Step 3

To write $\frac{5}{x - 2}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

$\frac{2 x}{x + 2} \cdot \frac{x - 2}{x - 2} + \frac{5}{x - 2} \cdot \frac{x + 2}{x + 2} - \frac{16}{(x + 2) (x - 2)}$

Step 4

Write each expression with a common denominator of $(x + 2) (x - 2)$ , by multiplying each by an appropriate factor of $1$ .

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

$\frac{2 x (x - 2)}{(x + 2) (x - 2)} + \frac{5}{x - 2} \cdot \frac{x + 2}{x + 2} - \frac{16}{(x + 2) (x - 2)}$

Multiply $\frac{5}{x - 2}$ by $\frac{x + 2}{x + 2}$ .

$\frac{2 x (x - 2)}{(x + 2) (x - 2)} + \frac{5 (x + 2)}{(x - 2) (x + 2)} - \frac{16}{(x + 2) (x - 2)}$

Reorder the factors of $(x - 2) (x + 2)$ .

$\frac{2 x (x - 2)}{(x + 2) (x - 2)} + \frac{5 (x + 2)}{(x + 2) (x - 2)} - \frac{16}{(x + 2) (x - 2)}$

Step 5

Simplify terms.

Combine the numerators over the common denominator.

$\frac{2 x (x - 2) + 5 (x + 2)}{(x + 2) (x - 2)} - \frac{16}{(x + 2) (x - 2)}$

Combine the numerators over the common denominator.

$\frac{2 x (x - 2) + 5 (x + 2) - 16}{(x + 2) (x - 2)}$

Step 6

Simplify each term.

Apply the distributive property.

$\frac{2 x \cdot x + 2 x \cdot - 2 + 5 (x + 2) - 16}{(x + 2) (x - 2)}$

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

Move $x$ .

$\frac{2 (x \cdot x) + 2 x \cdot - 2 + 5 (x + 2) - 16}{(x + 2) (x - 2)}$

Multiply $x$ by $x$ .

$\frac{2 x^{2} + 2 x \cdot - 2 + 5 (x + 2) - 16}{(x + 2) (x - 2)}$

Multiply $- 2$ by $2$ .

$\frac{2 x^{2} - 4 x + 5 (x + 2) - 16}{(x + 2) (x - 2)}$

Apply the distributive property.

$\frac{2 x^{2} - 4 x + 5 x + 5 \cdot 2 - 16}{(x + 2) (x - 2)}$

Multiply $5$ by $2$ .

$\frac{2 x^{2} - 4 x + 5 x + 10 - 16}{(x + 2) (x - 2)}$

Step 7

Simplify by adding terms.

Add $- 4 x$ and $5 x$ .

$\frac{2 x^{2} + x + 10 - 16}{(x + 2) (x - 2)}$

Subtract $16$ from $10$ .

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

Step 8

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 6 = - 12$ and whose sum is $b = 1$ .

Multiply by $1$ .

$\frac{2 x^{2} + 1 x - 6}{(x + 2) (x - 2)}$

Rewrite $1$ as $- 3$ plus $4$

$\frac{2 x^{2} + (- 3 + 4) x - 6}{(x + 2) (x - 2)}$

Apply the distributive property.

$\frac{2 x^{2} - 3 x + 4 x - 6}{(x + 2) (x - 2)}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(2 x^{2} - 3 x) + 4 x - 6}{(x + 2) (x - 2)}$

Factor out the greatest common factor (GCF) from each group.

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

Factor the polynomial by factoring out the greatest common factor, $2 x - 3$ .

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

Step 9

Cancel the common factor of $x + 2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify (2x)/(x+2)+5/(x-2)-16/(x^2-4)

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