The number of solutions of `z^2+overlinez = 0` is

Options

A. 1
B. 2
C. 3
D. 4

Answer

D

Explanation

Let `z=x+iy`. Then,
`z^(2)+bar(z)=0`
`rArr (x^(2)-y^(2)+2ixy)+(x-iy)=0`
`rArr x^(2)-y^(2)+x=0` and `2xy-y=0`
Now, `2xy-y=0 rArr y(2x-1)=0 rArr y=0` or `x=1//2`
If y=0, then
`x^(2)-y^(2)+x rArr 0 rArr x^(2)+x=0 rArr x=0` or `x=-1`
If `x=1//2`, then
`x^(2)-y^(2)+x=0 rArr y=+-sqrt(3)//2`
Thus, the given equation has four solutions.