Simplify 3/(x^2-9)+5/(x+3)

$\frac{3}{x^{2} - 9} + \frac{5}{x + 3}$

Step 1

Simplify the denominator.

Rewrite $9$ as $3^{2}$ .

$\frac{3}{x^{2} - 3^{2}} + \frac{5}{x + 3}$

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 $b = 3$ .

$\frac{3}{(x + 3) (x - 3)} + \frac{5}{x + 3}$

Step 2

To write $\frac{5}{x + 3}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{3}{(x + 3) (x - 3)} + \frac{5}{x + 3} \cdot \frac{x - 3}{x - 3}$

Step 3

Simplify terms.

Multiply $\frac{5}{x + 3}$ by $\frac{x - 3}{x - 3}$ .

$\frac{3}{(x + 3) (x - 3)} + \frac{5 (x - 3)}{(x + 3) (x - 3)}$

Combine the numerators over the common denominator.

$\frac{3 + 5 (x - 3)}{(x + 3) (x - 3)}$

Step 4

Simplify the numerator.

Apply the distributive property.

$\frac{3 + 5 x + 5 \cdot - 3}{(x + 3) (x - 3)}$

Multiply $5$ by $- 3$ .

$\frac{3 + 5 x - 15}{(x + 3) (x - 3)}$

Subtract $15$ from $3$ .

$\frac{5 x - 12}{(x + 3) (x - 3)}$