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

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

Simplify the denominator.

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

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

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 $b = 3$ .

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

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

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

Write each expression with a common denominator of $(x + 3) (x - 3)$ , by multiplying each by an appropriate factor of $1$ .

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

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

Reorder the factors of $(x - 3) (x + 3)$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $- 3$ by $3$ .

$\frac{2 x - 3 x - 9}{(x + 3) (x - 3)}$

Subtract $3 x$ from $2 x$ .

$\frac{- x - 9}{(x + 3) (x - 3)}$

Simplify with factoring out.

Factor $- 1$ out of $- x$ .

$\frac{- (x) - 9}{(x + 3) (x - 3)}$

Rewrite $- 9$ as $- 1 (9)$ .

$\frac{- (x) - 1 (9)}{(x + 3) (x - 3)}$

Factor $- 1$ out of $- (x) - 1 (9)$ .

$\frac{- (x + 9)}{(x + 3) (x - 3)}$

Simplify the expression.

Rewrite $- (x + 9)$ as $- 1 (x + 9)$ .

$\frac{- 1 (x + 9)}{(x + 3) (x - 3)}$

Move the negative in front of the fraction.

$- \frac{x + 9}{(x + 3) (x - 3)}$