Find the roots of: $$ z^2 -3z +4iz = 1-5i $$
Rearranging the terms: $z^2 + z(4i-3) + 5i - 1 $
Solving by using the quadratic formula:
$$z_{1,2} = \frac{3-4i\pm \sqrt{(4i-3)^2 -4(5i-1)}}{2}$$
If I simplify the root, I get something relatively ugly. Is that what I am supposed to do?
What you are doing is correct. You have $$ \begin {align*} \sqrt {(4i-3)^2 - 4(5i-1)} &= \sqrt {-16 + 9 - 24i - 20i + 4} \\&= \sqrt {-3 - 44i}. \end {align*} $$From here, you can use DeMoivre's theorem and it is indeed ugly. Nothing you can do if that's the question.