Answer
$\left\{ -\dfrac{2}{3}, \dfrac{2}{3}, -i, i \right\}$
Work Step by Step
The 2 numbers whose product is $ac=
9(-4)=-36
$ and whose sum is $b=
5
$ are $\{
9,-4
.\}$ Using these two numbers to decompose the middle term of the given equation, $
9x^4+5x^2-4=0
,$ then the factored form is
\begin{array}{l}\require{cancel}
9x^4+9x^2-4x^2-4=0
\\\\
(9x^4+9x^2)-(4x^2+4)=0
\\\\
9x^2(x^2+1)-4(x^2+1)=0
\\\\
(x^2+1)(9x^2-4)=0
.\end{array}
Equating each factor to zero, then,
\begin{array}{l}\require{cancel}
x^2+1=0
\\\\
x^2=-1
\\\\
x=\pm\sqrt{-1}
\\\\
x=\pm i
,\\\\\text{OR}\\\\
9x^2-4=0
\\\\
9x^2=4
\\\\
x^2=\dfrac{4}{9}
\\\\
x=\pm\sqrt{\dfrac{4}{9}}
\\\\
x=\pm\sqrt{\left( \dfrac{2}{3} \right)^2}
\\\\
x=\pm\dfrac{2}{3}
.\end{array}
Hence, the solutions are $
\left\{ -\dfrac{2}{3}, \dfrac{2}{3}, -i, i \right\}
$.
