# Complex Analysis curve

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
You could write out the equation with e.g. $a = Re(a) + Im(a) i$, so that after substituting $x = Re(z)$ and $y = Im(z)$ you get something of the form $A x^2 + B x y + C y^2 + D x + E y + F = 0$, with $A,B,C,D,E,F \in \mathbb{R}$... –  TMM Sep 11 '11 at 21:02

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 $$az^2+bz+c=a_1(x^2-y^2)-2a_2xy+b_1x-b_2y+c_1+i\cdot(\text{something real}),$$ and you are done.

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