Answer
$\left\{ -\dfrac{1}{2}, \dfrac{1}{2}, -i\sqrt{3}, i\sqrt{3} \right\}$
Work Step by Step
Using the properties of equality, the given equation, $
4x^4+11x^2=3
,$ is equivalent to
\begin{array}{l}\require{cancel}
4x^4+11x^2-3=0
.\end{array}
The 2 numbers whose product is $ac=
4(-3)=-12
$ and whose sum is $b=
11
$ are $\{
12,-1
.\}$ Using these two numbers to decompose the middle term of the expression, $
4x^4+11x^2-3=0
,$ then the factored form is
\begin{array}{l}\require{cancel}
4x^4+12x^2-x^2-3=0
\\\\
(4x^4+12x^2)-(x^2+3)=0
\\\\
4x^2(x^2+3)-(x^2+3)=0
\\\\
(x^2+3)(4x^2-1)=0
.\end{array}
Equating each factor to zero, then,
\begin{array}{l}\require{cancel}
x^2+3=0
\\\\
x^2=-3
\\\\
x=\pm\sqrt{-3}
\\\\
x=\pm i\sqrt{3}
,\\\\\text{OR}\\\\
4x^2-1=0
\\\\
4x^2=1
\\\\
x^2=\dfrac{1}{4}
\\\\
x=\pm\sqrt{\dfrac{1}{4}}
\\\\
x=\pm\sqrt{\left( \dfrac{1}{2} \right)^2}
\\\\
x=\pm\dfrac{1}{2}
.\end{array}
Hence, the solutions are $
\left\{ -\dfrac{1}{2}, \dfrac{1}{2}, -i\sqrt{3}, i\sqrt{3} \right\}
$.
