Question

Let
S = \left\\{z∈C : z^2+\overline{z} = 0 \right\\}. Then $∑_{z∈S}(Re(z)+Im(z))$
is equal to____.

Answer 1

The correct answer is 0
$∵ z^2+\overline{z} = 0$ Let $z = x+iy$
$∴ x^2+y^2+2ixy+x-iy = 0$
$(x^2-y^2+x)+i(2xy-y) = 0$
$∴ x^2+y^2 = 0$ and $(2x-1)y = 0$
If $x = +\frac{1}{2}$ then $y = ±\frac{\sqrt3}{2}$
And if y = 0 then x = 0, –1
$∴$ $z = \\{ 0 + 0i, -1 + 0i, \frac{1}{2} + \frac{\sqrt{3}}{2}i, \frac{1}{2} - \frac{\sqrt{3}}{2}i \\}$
$∴ ∑(R_e(z)+m(z)) = 0$