Answer
$-\frac{1}{2}\pm\frac{\sqrt{7}}{2}i$
Work Step by Step
$z^{2}+z+2=0$
We solve using the quadratic formula (a=1, b=1, c=2):
$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
$z=\displaystyle \frac{-1\pm\sqrt{1^{2}-4(1)(2)}}{2(1)}$
$=\frac{-1\pm\sqrt{1-8}}{2}$
$=\frac{-1\pm\sqrt{-7}}{2}$
$=\frac{-1\pm\sqrt{-1*7}}{2}$
We use the fact that $\sqrt{-1}=i$:
$=-\frac{1}{2}\pm\frac{\sqrt{7}}{2}i$