Question

The number of the solution of the equation $z^2+\overline{z}=0$ is

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is D $4$

Let $z=x+iy$, then
$z^2+\overline{z}=0$
$\Rightarrow (x^2-y^2+2ixy)+(x-iy)=0$
$\Rightarrow x^2-y^2+x=0$ and $2xy-y=0$
Now, $2xy-y=0\Rightarrow y(2x-1)=0$
$\Rightarrow y=0$ or $x=\frac{1}{2}$
If $y=0$, then
$x^2-y^2+x=0\Rightarrow x^2+x=0$
$\Rightarrow x=0$ or $x=-1$
If $x=\frac{1}{2}$, then $x^2-y^2+x=0$
$\Rightarrow y=\pm \frac{\sqrt{3}}{2}$
Thus, the given equations has four solutions.