Answer
$x=\left\{ -2i\sqrt{2},2i\sqrt{2} \right\}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
3x^2+24=0
,$ express first in the form $x^2=c.$ Then take the square root of both sides. Use the properties of radicals and the definition of an imaginary number to simplify the resulting radical
$\bf{\text{Solution Details:}}$
Using the properties of equality, the given equation is equivalent to
\begin{array}{l}\require{cancel}
3x^2=-24
\\\\
x^2=-\dfrac{24}{3}
\\\\
x^2=-8
.\end{array}
Taking the square root of both sides (Square Root Property), the equation above is equivalent to
\begin{array}{l}\require{cancel}
x=\pm\sqrt{-8}
.\end{array}
Using the Product Rule of radicals which is given by $\sqrt[m]{x}\cdot\sqrt[m]{y}=\sqrt[m]{xy},$ the equaion above is equivalent to
\begin{array}{l}\require{cancel}
x=\pm\sqrt{-1}\cdot\sqrt{8}
.\end{array}
Using the definition of an imaginary number which is given by $i=\sqrt{-1},$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
x=\pm i\sqrt{8}
.\end{array}
Extracting the root of the factor that is a perfect power of the index results to
\begin{array}{l}\require{cancel}
x=\pm i\sqrt{4\cdot2}
\\\\
x=\pm i\sqrt{(2)^2\cdot2}
\\\\
x=\pm i\cdot2\sqrt{2}
\\\\
x=\pm 2i\sqrt{2}
.\end{array}
Hence the solutions are $
x=\left\{ -2i\sqrt{2},2i\sqrt{2} \right\}
.$
