This is not really a calculus question but I am taking a course in basic complex analysis and this is from the first chapter.

Obviously you can plug $\displaystyle z_1$ in and get a true statement, but how do you go about finding an additional solution?

Show that the indicated numbers satisfy the given equation and find an additional solution, $\displaystyle z_2$.

$\displaystyle z^2+i=0, z_1=-\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i$