deletions | additions       diff --git a/untitled.tex b/untitled.tex  index 7e1778d..5f0aa5e 100644  --- a/untitled.tex  +++ b/untitled.tex 
...
Then we have, $\left|z^2\right|+Re(az)+b=0,$\\  By substitution we get the following: $x^2+y^2 + Re((c+di)(x+yi))+b =0$ \\  By algebra, $$x^2+y^2 + Re(cx+cyi+dxi+dyi^2)+b=0$$\\ Re(cx+cyi+dxi+dyi^2)=-b$$\\  $$ =x^2+y^2 x^2+y^2  + Re(cx-dy+i(dx+cy))+b=0$$\\  $\Leftrightarrow Re(cx-dy+i(dx+cy))=-b$$\\  $$  x^2 +y^2 + cx-dy + b=0$\\ b=0$$\\  $\Leftrightarrow (x^2+cx) +(y^2-dy) +b =0$\\  $\Leftrightarrow (x+\frac{1}{2}c)^2 + (y-\frac{1}{2}d)^2 = -b + \frac{1}{4}(c^2+d^2)$ by completing the square\\  The equation above is a circle with a radius of $\sqrt{-b + \frac{1}{4}(c^2+d^2)}.$\\