Question

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

• A. $2$
• B. $4$
• C. $6$
• D. $8$

Answer 1

The correct option is B $4$

Let $z=x+iy$ then $z^2+\overline{z}=0$
$\Rightarrow x^2-y^2+2ixy+x-iy=0$
$\Rightarrow x^2-y^2+x+2ixy-iy=0$
$\Rightarrow x^2-y^2+x=0;y(2x-1)=0$
$\Rightarrow y=0orx=1/2$
if $y=0$
$x^2+x=0\Rightarrow x(x+1)=0\Rightarrow x=0,x=-1$
if $x=1/2$
$\frac{1}{4}+\frac{1}{2}-y^2=0\Rightarrow \frac{3}{4}=y^2\Rightarrow y=\pm \frac{\sqrt{3}}{2}$
$\therefore 0+0i,-1+0i,\frac{1}{2}+\frac{\sqrt{3}}{2}i,\frac{1}{2}-\frac{\sqrt{3}}{2}i$ are four solutions