Difficult exam question - finding complex numbers from a quadratic.

Question

This appeared in a past exam paper for the complex calculus course I am taking. I am required to find all solutions for it and plot them in the complex plane. Below is my working.

I stopped before calculating $z$ because it clearly won't get nicer. I did not try applying De Moivre's Theorem because the argument for $-3+4i$ does not seem to have a simple form. I am pretty sure I have gone wrong somewhere or am using the wrong technique to find solutions because this question is not worth many points and the exam is only two hours long so I should be able to work it out quite quickly. How would I go about doing this kind of question quickly? Thank you.

Answer 1

You made an error solving the quartic: $(-3)^2-4(-4)=25$ (not $17$). Note that $$(1+2i)^2=1+(2i)^2+2(2i)=1-4+4i=-3+4i.$$ Therefore, according to your work $$z_1=\frac{3+(1+2i)}{2}=2+i\quad\text{and}\quad z_2=\frac{3-(1+2i)}{2}=1-i.$$

Answer 2

You seem to have made a mistake in solving $$-y^4+3y^2+4 =0$$

which should yield $y^2 = 4$ and $y^2=-1$.