Here's what I've done so far.

$\displaystyle \displaystyle \begin{align*} z^2 = -8 + 6i \\(x + yi)^2 = -8 + 6i \\ (x^2 - y^2) + (2xy)i = -8 + 6i \\ \\ x^2 - y^2 = -8 \textrm{ and } 2xy = 6 \end{align*} $

Using the second equation:

$\displaystyle \displaystyle \begin{align*} x = \frac{3}{y} \end{align*} $

Substituting into the first equation:

$\displaystyle \displaystyle \begin{align*} \frac{9}{y^2} - y^2 = -8 \\ 9 - y^4 = -8y^2 \\ -y^4 + 8y^2 + 9 = 0 \end{align*} $

And that is where I'm stuck, I thought of using a horner scheme but still didn't make it

P.S: Could someone indicate how to correctly set the alignment of the last equation ?