Simplify (3x)/(x^2-4)+6/(x+2)

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

Step 1

Simplify the denominator.

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

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

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 $b = 2$ .

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

Step 2

To write $\frac{6}{x + 2}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

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

Step 3

Simplify terms.

Multiply $\frac{6}{x + 2}$ by $\frac{x - 2}{x - 2}$ .

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

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

Factor $3$ out of $3 x + 6 (x - 2)$ .

Factor $3$ out of $3 x$ .

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

Factor $3$ out of $6 (x - 2)$ .

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

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

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

Apply the distributive property.

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

Multiply $2$ by $- 2$ .

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

Add $x$ and $2 x$ .

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