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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+i)(z−i)=−i:

Multiply both sides by z−i,

z+i=−i(z−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+i)(z−i)=i:

Multiply both sides by z−i,

z+i=i(z−i).

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

\displaystyle z-iz=1-i\,\textrm{.}

This gives

\displaystyle z = \frac{1-i}{1-i} = 1\,\textrm{.}

The solutions are therefore \displaystyle z=-1 and \displaystyle z=1\,.