## Algebra: A Combined Approach (4th Edition)

$x=\left\{ -2i\sqrt{2},2i\sqrt{2} \right\}$
$\bf{\text{Solution Outline:}}$ 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 $x=\left\{ -2i\sqrt{2},2i\sqrt{2} \right\} .$