a)$$ z(4+i)=i \\z=\frac{i}{4+i}\frac{4-i}{4-i}=\frac{4i+1}{17}=\frac{1}{17}+\frac{4}{17}i $$
b)$$(x+yi)^2+i=0\\x^2-y^2+2xyi=-i$$Koeffizientenvergleich:$$2xyi=-i \Rightarrow x=-\frac{1}{2y}\\(-\frac{1}{2y})^2-y^2=0 \Rightarrow y_{1,2}=\pm \frac{1}{\sqrt{2}}$$$$z_{1}=-\frac{\sqrt{2}}{2}+\frac{1}{\sqrt{2}}i \quad\wedge \quad z_2=\frac{\sqrt{2}}{2}-\frac{1}{\sqrt{2}}i\\$$
c) $$(x+yi)(x-yi)=4 \wedge (x+yi)+(x−yi)=0\\x^2+y^2=4 \wedge 2x=0\\x=0 \wedge y_{1,2}=\pm2\\z_{1,2}=\pm2i $$
