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

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

Step 1

Simplify the denominator.

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

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

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 $b = 3$ .

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

Step 2

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

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

Step 3

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

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

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

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

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

Apply the distributive property.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $3$ by $3$ .

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