## Intermediate Algebra (12th Edition)

$x=\left\{ \dfrac{2-i}{3},\dfrac{2+i}{3} \right\}$
$\bf{\text{Solution Outline:}}$ To find the solutions of the given equation, $(x-1)(9x-3)=-2 ,$ express first in the form $ax^2+bx+c=0.$ Then use the Quadratic Formula. $\bf{\text{Solution Details:}}$ Using the FOIL Method which is given by $(a+b)(c+d)=ac+ad+bc+bd,$ the expression above is equivalent to\begin{array}{l}\require{cancel} x(9x)+x(-3)-1(9x)-1(-3)=-2 \\\\ 9x^2-3x-9x+3=-2 \\\\ 9x^2+(-3x-9x)+(3+2)=0 \\\\ 9x^2-12x+5=0 .\end{array} The quadratic equation above has $a= 9 , b= -12 , c= 5 .$ Using $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$ or the Quadratic Formula, then \begin{array}{l}\require{cancel} x=\dfrac{-(-12)\pm\sqrt{(-12)^2-4(9)(5)}}{2(9)} \\\\ x=\dfrac{12\pm\sqrt{144-180}}{18} \\\\ x=\dfrac{12\pm\sqrt{-36}}{18} .\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} x=\dfrac{12\pm\sqrt{-1}\cdot\sqrt{36}}{18} .\end{array} Since $i=\sqrt{-1},$ the equation above is equivalent to \begin{array}{l}\require{cancel} x=\dfrac{12\pm i\sqrt{36}}{18} .\end{array} Simplifying the radicand by writing it as an expression that contains a factor that is a perfect square of the index and then extracting the root of that factor, the equation above is equivalent to \begin{array}{l}\require{cancel} x=\dfrac{12\pm i\sqrt{(6)^2}}{18} \\\\ x=\dfrac{12\pm 6i}{18} .\end{array} Cancelling the common factor in each term results to \begin{array}{l}\require{cancel} x=\dfrac{\cancel6(2)\pm \cancel6(1)i}{\cancel6(3)} \\\\ x=\dfrac{2\pm i}{3} .\end{array} The solutions are \begin{array}{l}\require{cancel} x=\dfrac{2-i}{3} \\\\\text{OR}\\\\ x=\dfrac{2+i}{3} .\end{array} Hence, $x=\left\{ \dfrac{2-i}{3},\dfrac{2+i}{3} \right\} .$