Answer
$\left\{-\dfrac{3}{2},4\right\}$
Work Step by Step
Multiplying both sides by the $LCD=
x^2
,$ the given equation, $
\dfrac{5}{x}+\dfrac{12}{x^2}=2
,$ is equivalent to
\begin{align*}
x^2\left(\dfrac{5}{x}+\dfrac{12}{x^2}\right)&=(2)x^2
\\\\
x(5) +1(12)&=2x^2
\\
5x+12&=2x^2
\\
0&=2x^2-5x-12
\\
2x^2-5x-12&=0
.\end{align*}
Using factoring of trinomials, the equation above is equivalent to
\begin{align*}
(x-4)(2x+3)&=0
.\end{align*}
Equating each factor to zero (Zero Product Property) and solving for the variable, then
\begin{array}{l|r}
x-4=0 & 2x+3=0
\\
x=4 & 2x=-3
\\\\
& x=-\dfrac{3}{2}
.\end{array}
Hence, the solution set of the equation $
\dfrac{5}{x}+\dfrac{12}{x^2}=2
$ is $\left\{-\dfrac{3}{2},4\right\}$.
