Quadratic equation and nth roots in C

Problem

Solve in $\mathbb{C}$:

(a) $z^{2}+2z+i=0$, and plot the solutions on the Gauss plane.

(b) Find all cube roots of $-2+2i$.

(c) Find all fifth roots of $32i$ and observe how the corresponding vectors are arranged.

Solution

(a) The discriminant is $\Delta=2^2-4i=4-4i$, so we need $\sqrt{4-4i}=2\sqrt{1-i}$. Writing $1-i=\sqrt2\,e^{-i\pi/4}$ gives $\sqrt{1-i}=2^{1/4}e^{-i\pi/8}\approx1.099-0.455i$. The solutions are therefore $z=-1\pm\sqrt{1-i}\approx\{0.099-0.455i,\ -2.099+0.455i\}.$ On the Gauss plane these correspond to two points symmetric about $(-1,0)$.

(Note: the answer given on the assignment states $z=-1\pm2i$, but that value solves the equation $z^2+2z+5=0$, not the assigned equation $z^2+2z+i=0$; the correct result for the given equation is the one computed above.)

(b) Write $-2+2i=2\sqrt2\,e^{i3\pi/4}$. The modulus of the cube roots is $(2\sqrt2)^{1/3}=\sqrt2$, and the arguments are $\dfrac{3\pi/4+2k\pi}3$ for $k=0,1,2$: $k=0:\ \sqrt2\,e^{i\pi/4}=1+i;\qquad k=1:\ \sqrt2\,e^{i11\pi/12};\qquad k=2:\ \sqrt2\,e^{i19\pi/12}.$ Indeed $(1+i)^3=-2+2i$, as required.

(c) Write $32i=32e^{i\pi/2}$. The modulus of the fifth roots is $32^{1/5}=2$, and the arguments are $\dfrac{\pi/2+2k\pi}5$ for $k=0,1,2,3,4$: $2e^{i\pi/10},\quad 2e^{i\pi/2}=2i,\quad 2e^{i9\pi/10},\quad 2e^{i13\pi/10},\quad 2e^{i17\pi/10}.$ The five corresponding vectors are the vertices of a regular pentagon centered at the origin.

$\boxed{z=-1\pm\sqrt{1-i};\qquad \sqrt[3]{-2+2i}=\{1+i,\ \sqrt2\,e^{i11\pi/12},\ \sqrt2\,e^{i19\pi/12}\};\qquad \sqrt[5]{32i}=\{2e^{i\pi/10},\,2i,\,2e^{i9\pi/10},\,2e^{i13\pi/10},\,2e^{i17\pi/10}\}}$