First slide

Algebra of complex numbers

Question

The number of complex numbers $z$ which satisfy $z^{2} + 2 | z |^{2} = 2$ is

Moderate

A

0
B

2
C

3
D

4

Solution

As $z^{2} = 2 (1 - | z |^{2})$ is real $z$ is either purely real or purely imaginary.
If $z$ is purely real, then
$z^{2} = 2 (1 - z^{2}) \Rightarrow z = \pm \sqrt{\frac{2}{3}}$
TIP
If z is purely imaginary, then $| z |^{2} = - z^{2} .$
in this case $z^{2} = 2 (1 + z^{2}) \Rightarrow z = \pm \sqrt{2} i$
Thus, there are four complex numbers satisfying
$z^{2} + 2 | z |^{2} = 2$

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