Answer
$\left \{-\frac{7}{2}\right \}$.
Work Step by Step
The given expression is
$=\frac{2}{x-2}+\frac{3}{x^2-4}=0$
$=\frac{2}{x-2}+\frac{3}{x^2-2^2}=0$
Use the special formula $A^2-B^2=(A-B)(A+B)$
$=\frac{2}{x-2}+\frac{3}{(x-2)(x+2)}=0$
The LCD is $(x-2)(x+2)$.
Multiply the equation by the LCD.
$=(x-2)(x+2)\cdot \frac{2}{x-2}+(x-2)(x+2)\cdot\frac{3}{(x-2)(x+2)}=0$
Cancel common terms.
$=2(x+2)+3=0$
Use the distributive property.
$=2x+4+3=0$
Simplify.
$=2x+7=0$
Subtract $7$ from both sides.
$=2x+7-7=0-7$
Simplify.
$=2x=-7$
Divide both sides by $2$.
$=\frac{2x}{2}=\frac{-7}{2}$
Simplify.
$=x=-\frac{7}{2}$
The solution set is $\left \{-\frac{7}{2}\right \}$.
