Answer
$\left\{-\dfrac{2}{3},6\right\}$
Work Step by Step
Multiplying both sides by the $LCD=
x^2
,$ the given equation, $
3-\dfrac{16}{x}-\dfrac{12}{x^2}=0
,$ is equivalent to
\begin{align*}
x^2\left(3-\dfrac{16}{x}-\dfrac{12}{x^2}\right)&=(0)x^2
\\\\
x^2(3)+x(-16)+1(-12)&=0
\\
3x^2-16x-12&=0
.\end{align*}
Using factoring of trinomials, the equation above is equivalent to
\begin{align*}
(x-6)(3x+2)&=0
.\end{align*}
Equating each factor to zero (Zero Product Property) and solving the variable, then
\begin{array}{l|r}
x-6=0 & 3x+2=0
\\
x=6 & 3x=-2
\\\\
& x=-\dfrac{2}{3}
.\end{array}
Hence, the solution set of the equation $
3-\dfrac{16}{x}-\dfrac{12}{x^2}=0
$ is $\left\{-\dfrac{2}{3},6\right\}$.
