Answer
$x = 0.343+1.455~i$
$x = -0.343-1.455~i$
Work Step by Step
$x^2+2-i = 0$
$x^2 = -2+i$
We can find $r$:
$r = \sqrt{(-2)^2+(1)^2} = \sqrt{5}$
Note that $~~-2+i~~$ is in the second quadrant:
$\theta = tan^{-1}(-\frac{1}{2})+180^{\circ}$
$\theta = -26.565^{\circ}+180^{\circ} = 153.435^{\circ}$
We can find $x$:
When $k=0$:
$x = (\sqrt{5})^{1/2}(cos~\frac{153.435^{\circ}+360^{\circ}~k}{2}+i~sin~\frac{153.435^{\circ}+360^{\circ}~k}{2})$
$x = (\sqrt{5})^{1/2}(cos~76.72^{\circ}+i~sin~76.72^{\circ})$
$x = 0.343+1.455~i$
When $k=1$:
$x = (\sqrt{5})^{1/2}(cos~\frac{153.435^{\circ}+360^{\circ}~k}{2}+i~sin~\frac{153.435^{\circ}+360^{\circ}~k}{2})$
$x = (\sqrt{5})^{1/2}(cos~256.72^{\circ}+i~sin~256.72^{\circ})$
$x = -0.343-1.455~i$
