Question

f), g), h) de la exercițiul 9

Answer 1

f)
Notam t=(z+1)/(z-1) si obtinem t^3+t^2+t+1=0 <=> t^2(t+1)+(t+1)=0 <=> (t+1)(t^2+1)=0 <=> t=-1 sau t=-i sau t=i.

t=-1 => (z+1)/(z-1)=-1 <=> z+1=-z+1 <=> 2z=0 <=> z=0
t=-i => (z+1)/(z-1)=-i <=> z+1=-iz+i <=> z(i+1)=i-1 <=> z=(i-1)/(i+1)=(i-1)^2/(-2)=(-2i)/(-2)=i.
t=i => (z+1)/(z-1)=i <=> z+1=iz-i <=> (1-i)z=-1-i <=> z=-(1+i)/(1-i)=-(1+i)^2/2=-2i/2=-i.

g)
z^2-(1+i)z+i=0
delta=(-(1+i))^2-4i=2i-4i=-2i=(1-i)^2.
z1,2=(1+i+-(1-i))/2. => z1=1, z2=i.

h)
iz^2+(3+i)z+2-2i=0
delta=(3+i)^2-4i(2-2i)=8+6i-8i-8=-2i=(1-i)^2.
z1,2=(-3-i+-(1-i))/(2i). => z1=(-1-i)/i=i-1, z2=-2/i=2i