Answer
$x=\left\{ \dfrac{5-i\sqrt{55}}{2},\dfrac{5+i\sqrt{55}}{2}
\right\}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To find the solutions of the given equation, $
x^2-5x+20=0
,$ use the Quadratic Formula.
$\bf{\text{Solution Details:}}$
The quadratic equation above has $a=
1
, b=
-5
, c=
20
.$ Using $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$ or the Quadratic Formula, then
\begin{array}{l}\require{cancel}
x=\dfrac{-(-5)\pm\sqrt{(-5)^2-4(1)(20)}}{2(1)}
\\\\
x=\dfrac{5\pm\sqrt{25-80}}{2}
\\\\
x=\dfrac{5\pm\sqrt{-55}}{2}
.\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{5\pm\sqrt{-1}\cdot\sqrt{55}}{2}
.\end{array}
Since $i=\sqrt{-1},$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
x=\dfrac{5\pm i\sqrt{55}}{2}
.\end{array}
The solutions are
\begin{array}{l}\require{cancel}
x=\dfrac{5-i\sqrt{55}}{2}
\\\\\text{OR}\\\\
x=\dfrac{5+i\sqrt{55}}{2}
.\end{array}
Hence, $
x=\left\{ \dfrac{5-i\sqrt{55}}{2},\dfrac{5+i\sqrt{55}}{2}
\right\}
.$
