## Intermediate Algebra (12th Edition)

$x=\left\{ -4,8 \right\}$
$\bf{\text{Solution Outline:}}$ To solve the given equation, $m^2-4m-32=0 ,$ use first the properties of equality to express the equation in the form $x^2\pm bx=c.$ Once in this form, complete the square by adding $\left( \dfrac{b}{2} \right)^2$ to both sides of the equal sign. Then express the left side as a square of a binomial while simplify the right side. Then take the square root of both sides (Square Root Property) and use the properties of equality to isolate the variable. $\bf{\text{Solution Details:}}$ Using the properties of equality, in the form $x^2+bx=c,$ the given equation is equivalent to \begin{array}{l}\require{cancel} m^2-4m=32 .\end{array} In the equation above, $b= -4 .$ The expression $\left( \dfrac{b}{2} \right)^2,$ evaluates to \begin{array}{l}\require{cancel} \left( \dfrac{-4}{2} \right)^2 \\\\= \left( -2 \right)^2 \\\\= 4 .\end{array} Adding the value of $\left( \dfrac{b}{2} \right)^2,$ to both sides of the equation above results to \begin{array}{l}\require{cancel} m^2-4m+4=32+4 \\\\ m^2-4m+4=36 .\end{array} With the left side now a perfect square trinomial, the equation above is equivalent to \begin{array}{l}\require{cancel} (m-2)^2=36 .\end{array} Taking the square root of both sides (Square Root Property), simplifying the radical and then isolating the variable, the equation above is equivalent to \begin{array}{l}\require{cancel} m-2=\pm\sqrt{36} \\\\ m-2=\pm6 \\\\ m=2\pm6 .\end{array} The solutions are \begin{array}{l}\require{cancel} m=2-6 \\\\ m=-4 \\\\\text{OR}\\\\ m=2+6 \\\\ m=8 .\end{array} Hence, $x=\left\{ -4,8 \right\} .$