Let S=left{z in C : z^{2}+bar{z}=0right}. The | vedclass

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Question

$S=\left\{z \in C: z^{2}+\bar{z}=0\right\}$

Let $z = x + iy$

$z ^{2}= x ^{2}- y ^{2}+2 ixy$

$\bar{z}=x-i y$

$z^{2}+\bar{z}=x^{2}-y^{2}+x+i

Vedclass · Accepted Answer

$0$

Vedclass · Answer

$S=\left\{z \in C: z^{2}+\bar{z}=0ight\}$

Let $z = x + iy$

$z ^{2}= x ^{2}- y ^{2}+2 ixy$

$\bar{z}=x-i y$

$z^{2}+\bar{z}=x^{2}-y^{2}+x+i(2 x y-y)=0$

$x^{2}+x-y^{2}=0 	ext { \& } 2 x y-y=0$

$y=0$ or $x=\frac{1}{2}$

If $y =0 ; x =0,-1$

If $x=\frac{1}{2} ; y=\frac{\sqrt{3}}{2}, \frac{-\sqrt{3}}{2}$

$\sum_{z \in S}(\operatorname{Re}(z)+\operatorname{Im}(z)=\left(0-1+\frac{1}{2}+\frac{1}{2}ight)$

$+0+0+\frac{\sqrt{3}}{2}-\frac{\sqrt{3}}{2})$

Let $S=\left\{z \in C : z^{2}+\bar{z}=0\right\}$. Then $\sum \limits_{z \in S}(\operatorname{Re}(z)+\operatorname{Im}(z))$ is equal to$......$

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