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We are given three complex numbers a,b,and c. Consider $Re(az^{2} + bz +c)=0$. What is this curve? I am having a hard time approaching this problem. Any suggestions or help would be great.

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See this. – J. M. Sep 11 '11 at 21:01

If $a=a_1+ia_2$, $b=b_1+ib_2$, $c=c_1+ic_2$ and $z=x+iy$, then $$z^2=x^2-y^2+2ixy$$ hence $az^2+bz+c$ equals $$ a_1(x^2-y^2)-2a_2xy+b_1x-b_2y+c_1+i\cdot(\text{something real}), $$ and you are done.

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