Answer
$x=\left\{ \dfrac{3-\sqrt{73}}{2},\dfrac{3+\sqrt{73}}{2}
\right\}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To find the solutions of the given equation, $
(x-5)(x+2)=6
,$ 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(x)+x(2)-5(x)-5(2)=6
\\\\
x^2+2x-5x-10=6
\\\\
x^2-3x-10=6
.\end{array}
Using the properties of equality, in the form $ax^2+bx+c=0,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
x^2-3x-10-6=0
\\\\
x^2-3x-16=0
.\end{array}
The quadratic equation above has $a=
1
, b=
-3
, c=
-16
.$ Using $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$ or the Quadratic Formula, then
\begin{array}{l}\require{cancel}
x=\dfrac{-(-3)\pm\sqrt{(-3)^2-4(1)(-16)}}{2(1)}
\\\\
x=\dfrac{3\pm\sqrt{9+64}}{2}
\\\\
x=\dfrac{3\pm\sqrt{73}}{2}
.\end{array}
The solutions are
\begin{array}{l}\require{cancel}
x=\dfrac{3-\sqrt{73}}{2}
\\\\\text{OR}\\\\
x=\dfrac{3+\sqrt{73}}{2}
.\end{array}
Hence, $
x=\left\{ \dfrac{3-\sqrt{73}}{2},\dfrac{3+\sqrt{73}}{2}
\right\}
.$
