Answer
$\left\{\dfrac{-1-i\sqrt{3}}{2},\dfrac{-1+i\sqrt{3}}{2}\right\}$
Work Step by Step
The given equation, $
z^2+z+1=0
,$ has
\begin{align*}
a=
1
,\text{ }b=
1
,\text{ and }c=
1
.\end{align*}
Using $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$ or the Quadratic Formula, then
\begin{align*}\require{cancel}
x&=
\dfrac{-1\pm\sqrt{1^2-4(1)(1)}}{2(1)}
\\\\&=
\dfrac{-1\pm\sqrt{1-4}}{2}
\\\\&=
\dfrac{-1\pm\sqrt{-3}}{2}
\\\\&=
\dfrac{-1\pm\sqrt{3\cdot(-1)}}{2}
\\\\&=
\dfrac{-1\pm\sqrt{3}\cdot\sqrt{-1}}{2}
\\\\&=
\dfrac{-1\pm\sqrt{3}\cdot i}{2}
&(\text{use }i=\sqrt{-1})
\\\\&=
\dfrac{-1\pm i\sqrt{3}}{2}
.\end{align*}
Hence, the solution set of the equation $
z^2+z+1=0
$ is $\left\{\dfrac{-1-i\sqrt{3}}{2},\dfrac{-1+i\sqrt{3}}{2}\right\}$.
