Question

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

• A. $5$
• B. $4$
• C. $2$
• D. $3$

Answer 1

The correct option is B $4$

Let $z=x+iy$.

Now,

$z^2+\overline{z}=0$

or, $x^2-y^2+2ixy+(x-iy)=0$

or, $(x^2-y^2+x)+i(2xy-y)=0$

Now comparing the real and imaginary part both sides we get,

$x^2-y^2+x=0$.....(1) and $2xy-y=0$.....(2).

From (2) we get, $x=\frac{1}{2}$ or $y=0$.

Now $x=\frac{1}{2}$ gives from (1) we get, $y=\pm \frac{\sqrt{3}}{2}$.

And $y=0$ gives from (1) we get, $x=0,1$.

So the solution s are $(0,0),(1,0),(\frac{1}{2},\pm \frac{\sqrt{3}}{2})$.

So we have $4$ solutions.