Solving a complex quadratic-like equation Let $a,b,c,d$ are given non-zero complex numbers (i.e., constants). Then is it true that the equation 
$$a|z|^2+b z+c\bar{z}+d=0$$
which is equivalent to $$|z|^2+b'z+c'\bar{z}+d'=0$$
will always have a (at least one) solution for $z$? Or there is some necessary and sufficient condition that $a,b,c,d$ must satisfy to guarantee a solution in $z$? Notation: $\bar{z}$ is the complex conjugate of $z$.

Take $a=b=c=1,$ then $d = 47.$ With real $x,y$ and $z = x + iy,$ this is
$$  x^2 + y^2 + 2x + 47 = 0, $$
$$ (x+1)^2 + y^2 + 46 = 0 $$
