let $z = 1 +i$
Find all complex solutions such that $z^2 + \bar z^2 = 0$.
My working out:
$z^2 = -\bar z^2 = -(1-i)^2 = 2i$
so $z^2 = 2i$
hence $r^2 = 2 \implies r = \sqrt 2$
mod: $2\theta = \frac{\pi}{2} + 2k\pi \implies \theta = \frac{\pi}{4} + k(\pi)$ where $k = 0, 1$
overall roots are $z = \sqrt 2 \operatorname{cis} \left(\frac\pi4 +k\pi\right)$
Is my working out and solution correct?
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