## Intermediate Algebra (12th Edition)

$x=\left\{ \dfrac{7-3i\sqrt{3}}{4},\dfrac{7+ 3i\sqrt{3}}{4} \right\}$
$\bf{\text{Solution Outline:}}$ To find the non real complex solutions of the given equation, $(4m-7)^2=-27 ,$ take the square root of both sides (Square Root Property). Then use the properties of radicals and use $i=\sqrt{-1}.$ Finally, simplify the radical and isolate the variable. $\bf{\text{Solution Details:}}$ Taking the square root of both sides (Square Root Property), the equation above is equivalent to \begin{array}{l}\require{cancel} 4m-7=\pm\sqrt{-27} .\end{array} Using the Product Rule of radicals which is given by $\sqrt[m]{x}\cdot\sqrt[m]{y}=\sqrt[m]{xy},$ the equation above is equivalent to\begin{array}{l}\require{cancel} 4m-7=\pm\sqrt{-1}\cdot\sqrt{27} .\end{array} Using $i=\sqrt{-1},$ the equation above is equivalent to \begin{array}{l}\require{cancel} 4m-7=\pm i\sqrt{27} .\end{array} Writing the radicand as an expression that contains a factor that is a perfect power of the given index and then extracting the root of that factor, the equation above is equivalent to \begin{array}{l}\require{cancel} 4m-7=\pm i\sqrt{9\cdot3} \\\\ 4m-7=\pm i\sqrt{(3)^2\cdot3} \\\\ 4m-7=\pm i(3)\sqrt{3} \\\\ 4m-7=\pm 3i\sqrt{3} .\end{array} Using the properties of equality to isolate the variable, the equation above is equivalent to \begin{array}{l}\require{cancel} 4m=7\pm 3i\sqrt{3} \\\\ m=\dfrac{7\pm 3i\sqrt{3}}{4} .\end{array} Hence, $x=\left\{ \dfrac{7-3i\sqrt{3}}{4},\dfrac{7+ 3i\sqrt{3}}{4} \right\} .$