Answer
$x=\left\{ -\dfrac{14}{3},8 \right\}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
\log_4[(3x+8)(x-6)]=3
,$ change to exponential form. Then express the resulting equation in the form $ax^2+bx+c=0.$ Use the Quadratic Formula to solve for the values of the variable. Then do checking of the solutions with the original equation.
$\bf{\text{Solution Details:}}$
Since $y=b^x$ is equivalent to $\log_b y=x,$ the exponential form of the equation above is
\begin{array}{l}\require{cancel}
(3x+8)(x-6)=4^3
\\\\
(3x+8)(x-6)=64
.\end{array}
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}
3x(x)+3x(-6)+8(x)+8(-6)=64
\\\\
3x^2-18x+8x-48=64
\\\\
3x^2+(-18x+8x)+(-48-64)=0
\\\\
3x^2-10x-112=0
.\end{array}
In the equation above, $a=
3
,$ $b=
-10
,$ and $c=
-112
.$ Using the Quadratic Formula which is given by $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a},$ then
\begin{array}{l}\require{cancel}
x=\dfrac{-(-10)\pm\sqrt{(-10)^2-4(3)(-112)}}{2(3)}
\\\\
x=\dfrac{10\pm\sqrt{100+1344}}{6}
\\\\
x=\dfrac{10\pm\sqrt{1444}}{6}
\\\\
x=\dfrac{10\pm\sqrt{(38)^2}}{6}
\\\\
x=\dfrac{10\pm38}{6}
.\end{array}
The solutions are
\begin{array}{l}\require{cancel}
x=\dfrac{10-38}{6}
\\\\
x=\dfrac{-28}{6}
\\\\
x=\dfrac{-14}{3}
\\\\
x=-\dfrac{14}{3}
\\\\\text{OR}\\\\
x=\dfrac{10+38}{6}
\\\\
x=\dfrac{48}{6}
\\\\
x=8
.\end{array}
Upon checking, $
x=\left\{ -\dfrac{14}{3},8 \right\}
,$ satisfy the original equation.
