Simplify -3/(9x^2-4)+1/(3x^2+2x)

$\frac{- 3}{9 x^{2} - 4} + \frac{1}{3 x^{2} + 2 x}$

Step 1

Simplify each term.

Simplify the denominator.

Rewrite $9 x^{2}$ as ${(3 x)}^{2}$ .

$\frac{- 3}{{(3 x)}^{2} - 4} + \frac{1}{3 x^{2} + 2 x}$

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

$\frac{- 3}{{(3 x)}^{2} - 2^{2}} + \frac{1}{3 x^{2} + 2 x}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3 x$ and $b = 2$ .

$\frac{- 3}{(3 x + 2) (3 x - 2)} + \frac{1}{3 x^{2} + 2 x}$

Move the negative in front of the fraction.

$- \frac{3}{(3 x + 2) (3 x - 2)} + \frac{1}{3 x^{2} + 2 x}$

Factor $x$ out of $3 x^{2} + 2 x$ .

Factor $x$ out of $3 x^{2}$ .

$- \frac{3}{(3 x + 2) (3 x - 2)} + \frac{1}{x (3 x) + 2 x}$

Factor $x$ out of $2 x$ .

$- \frac{3}{(3 x + 2) (3 x - 2)} + \frac{1}{x (3 x) + x \cdot 2}$

Factor $x$ out of $x (3 x) + x \cdot 2$ .

$- \frac{3}{(3 x + 2) (3 x - 2)} + \frac{1}{x (3 x + 2)}$

Step 2

To write $- \frac{3}{(3 x + 2) (3 x - 2)}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$- \frac{3}{(3 x + 2) (3 x - 2)} \cdot \frac{x}{x} + \frac{1}{x (3 x + 2)}$

Step 3

To write $\frac{1}{x (3 x + 2)}$ as a fraction with a common denominator, multiply by $\frac{3 x - 2}{3 x - 2}$ .

$- \frac{3}{(3 x + 2) (3 x - 2)} \cdot \frac{x}{x} + \frac{1}{x (3 x + 2)} \cdot \frac{3 x - 2}{3 x - 2}$

Step 4

Write each expression with a common denominator of $(3 x + 2) (3 x - 2) x$ , by multiplying each by an appropriate factor of $1$ .

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

$- \frac{3 x}{(3 x + 2) (3 x - 2) x} + \frac{1}{x (3 x + 2)} \cdot \frac{3 x - 2}{3 x - 2}$

Multiply $\frac{1}{x (3 x + 2)}$ by $\frac{3 x - 2}{3 x - 2}$ .

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

Reorder the factors of $(3 x + 2) (3 x - 2) x$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

Add $- 3 x$ and $3 x$ .

$\frac{0 - 2}{x (3 x + 2) (3 x - 2)}$

Subtract $2$ from $0$ .

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

Step 7

Move the negative in front of the fraction.

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