Lösung 3.3:2e

Aus Online Mathematik Brückenkurs 2

If we treat the expression w=z−iz+i as an unknown, we have the equation

w2=−1

We know already that this equation has roots

w=−ii 

so z should satisfy one of the equation's

z−iz+i=−i or z−iz+i=i

We solve these equations one by one.

z−iz+i=−i

Multiply both sides by z−i:

z+i=−iz−i 

Move all the z -terms over to the left-hand side and all the constants to the right-hand side,

z+iz=−1−i

This gives

z=1+i−1−i=1+i−1+i=−1 

z−iz+i=i

Multiply both sides by z−i:

z+i=iz−i 

Move all the z-terms over to the left-hand side and all the constants to the right-hand side,

z−iz=1−i

This gives

z=1−i1−i=1

The solutions are therefore z=−1 and z=1.