\documentclass[a4paper]{article}
\begin{document}
\begin{eqnarray*}
f (z)
 & = & \frac{1}{az^{2} + bz + c} \\
D
 & = & b^{2} - 4ac \, \, \, = \, \, \, 0 \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
f (z)
 & = & \frac{1}{az^{2} + bz + c} \\
& = & \frac{1}{a (x + iy)^{2} + b (x + iy) + c} \\
& = & \frac{1}{ax^{2} + 2iaxy - ay^{2} + bx + iby + c} \\
& = & \frac{1}{(ax^{2} - ay^{2} + bx + c) + i (2axy + by)} \\
& = & \frac{1}{(ax^{2} - ay^{2} + bx + c) + i (2axy + by)} \frac{(ax^{2} - ay^{2} + bx + c) - i (2axy + by)}{(ax^{2} - ay^{2} + bx + c) - i (2axy + by)} \\
& = & \frac{(ax^{2} - ay^{2} + bx + c) - i (2axy + by)}{(ax^{2} - ay^{2} + bx + c)^{2} + (2axy + by)^{2}} \\
& = & \underbrace{\frac{ax^{2} - ay^{2} + bx + c}{(ax^{2} - ay^{2} + bx + c)^{2} + (2axy + by)^{2}}}_{u}
 + \, i \underbrace{\frac{- 2axy - by}{(ax^{2} - ay^{2} + bx + c)^{2} + (2axy + by)^{2}}}_{v} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
N
 & = & \mbox{Noemer} \\
& = & (ax^{2} - ay^{2} + bx + c)^{2} + (2axy + by)^{2} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\frac{\partial u}{\partial x}
 & = & \frac{\partial }{\partial x} \left ( \frac{ax^{2} - ay^{2} + bx + c}{(ax^{2} - ay^{2} + bx + c)^{2} + (2axy + by)^{2}} \right ) \\
& = & ((2ax + b) ((ax^{2} - ay^{2} + bx + c)^{2} + (2axy + by)^{2}) \\
 & & \qquad - \, (ax^{2} - ay^{2} + bx + c) (2 (ax^{2} - ay^{2} + bx + c) (2ax + b) + 2 (2axy + by) (2ay))) \frac{1}{N^{2}} \\
& = & (2ax + b) (((ax^{2} - ay^{2} + bx + c)^{2} + y^{2} (2ax + b)^{2}) \\
 & & \qquad - \, (ax^{2} - ay^{2} + bx + c) (2 (ax^{2} - ay^{2} + bx + c) + 4ay^{2})) \frac{1}{N^{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\frac{\partial u}{\partial y}
 & = & \frac{\partial }{\partial y} \left ( \frac{ax^{2} - ay^{2} + bx + c}{(ax^{2} - ay^{2} + bx + c)^{2} + (2axy + by)^{2}} \right ) \\
& = & ((- 2ay) ((ax^{2} - ay^{2} + bx + c)^{2} + (2axy + by)^{2}) \\
 & & \qquad - \, (ax^{2} - ay^{2} + bx + c) (2 (ax^{2} - ay^{2} + bx + c) (- 2ay) + 2 (2axy + by) (2ax + b))) \frac{1}{N^{2}} \\
& = & ((- 2ay) ((ax^{2} - ay^{2} + bx + c)^{2} + y^{2} (2ax + b)^{2}) \\
 & & \qquad - \, (ax^{2} - ay^{2} + bx + c) ((- 4ay) (ax^{2} - ay^{2} + bx + c) + 2y (2ax + b)^{2})) \frac{1}{N^{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\frac{\partial v}{\partial x}
 & = & \frac{\partial }{\partial x} \left ( \frac{- 2axy - by}{(ax^{2} - ay^{2} + bx + c)^{2} + (2axy + by)^{2}} \right ) \\
& = & ((- 2ay) ((ax^{2} - ay^{2} + bx + c)^{2} + (2axy + by)^{2}) \\
 & & \qquad - \, (- 2axy - by) (2 (ax^{2} - ay^{2} + bx + c) (2ax + b) + 2 (2axy + by) (2ay))) \frac{1}{N^{2}} \\
& = & ((- 2ay) ((ax^{2} - ay^{2} + bx + c)^{2} + y^{2} (2ax + b)^{2}) \\
 & & \qquad + \, y (2ax + b)^{2} (2 (ax^{2} - ay^{2} + bx + c) + 4ay^{2})) \frac{1}{N^{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\frac{\partial v}{\partial y}
 & = & \frac{\partial }{\partial y} \left ( \frac{- 2axy - by}{(ax^{2} - ay^{2} + bx + c)^{2} + (2axy + by)^{2}} \right ) \\
& = & ((- 2ax - b) ((ax^{2} - ay^{2} + bx + c)^{2} + (2axy + by)^{2}) \\
 & & \qquad - \, (- 2axy - by) (2 (ax^{2} - ay^{2} + bx + c) (- 2ay) + 2 (2axy + by) (2ax + b))) \frac{1}{N^{2}} \\
& = & (- (2ax + b) ((ax^{2} - ay^{2} + bx + c)^{2} + y^{2} (2ax + b)^{2}) \\
 & & \qquad + \, y (2ax + b) ((- 4ay) (ax^{2} - ay^{2} + bx + c) + 2y (2ax + b)^{2})) \frac{1}{N^{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\frac{\partial u}{\partial x} - \frac{\partial v}{\partial y}
 & = & 0 \quad \wedge \quad \frac{\partial v}{\partial x} + \frac{\partial u}{\partial y} \, \, \, = \, \, \, 0 \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\frac{\partial u}{\partial x} - \, \frac{\partial v}{\partial y}
 & = & (2ax + b) (((ax^{2} - ay^{2} + bx + c)^{2} + y^{2} (2ax + b)^{2}) \\
 & & \qquad - \, (ax^{2} - ay^{2} + bx + c) (2 (ax^{2} - ay^{2} + bx + c) + 4ay^{2})) \frac{1}{N^{2}} \\
 & & \qquad \qquad - \, (- (2ax + b) ((ax^{2} - ay^{2} + bx + c)^{2} + y^{2} (2ax + b)^{2}) \\
 & & \qquad \qquad \qquad + \, y (2ax + b) ((- 4ay) (ax^{2} - ay^{2} + bx + c) + 2y (2ax + b)^{2})) \frac{1}{N^{2}} \\
& = & ((2ax + b) (ax^{2} - ay^{2} + bx + c)^{2} + y^{2} (2ax + b)^{3} \\
 & & \qquad - \, 2 (ax^{2} - ay^{2} + bx + c)^{2} (2ax + b) - 4ay^{2} (ax^{2} - ay^{2} + bx + c) (2ax + b) \\
 & & \qquad \qquad + \, (2ax + b) (ax^{2} - ay^{2} + bx + c)^{2} + y^{2} (2ax + b)^{3} \\
 & & \qquad \qquad \qquad + \, 4ay^{2} (2ax + b) (ax^{2} - ay^{2} + bx + c) - 2y^{2} (2ax + b)^{3}) \frac{1}{N^{2}} \\
& = & 0 \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\frac{\partial u}{\partial y} + \, \frac{\partial v}{\partial x}
 & = & ((- 2ay) ((ax^{2} - ay^{2} + bx + c)^{2} + y^{2} (2ax + b)^{2}) \\
 & & \qquad - \, (ax^{2} - ay^{2} + bx + c) ((- 4ay) (ax^{2} - ay^{2} + bx + c) + 2y (2ax + b)^{2})) \frac{1}{N^{2}} \\
 & & \qquad \qquad + \, ((- 2ay) ((ax^{2} - ay^{2} + bx + c)^{2} + y^{2} (2ax + b)^{2}) \\
 & & \qquad \qquad \qquad + \, y (2ax + b)^{2} (2 (ax^{2} - ay^{2} + bx + c) + 4ay^{2})) \frac{1}{N^{2}} \\
& = & (- 2ay (ax^{2} - ay^{2} + bx + c)^{2} - 2ay^{3} (2ax + b)^{2} \\
 & & \qquad + \, 4ay (ax^{2} - ay^{2} + bx + c)^{2} - 2y (2ax + b)^{2} (ax^{2} - ay^{2} + bx + c) \\
 & & \qquad \qquad - \, 2ay (ax^{2} - ay^{2} + bx + c)^{2} - 2ay^{3} (2ax + b)^{2} \\
 & & \qquad \qquad \qquad + \, 2y (2ax + b)^{2} (ax^{2} - ay^{2} + bx + c) + 4ay^{3} (2ax + b)^{2}) \frac{1}{N^{2}} \\
& = & 0 \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
P
 & = & 2ax + b \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
Q
 & = & ax^{2} - ay^{2} + bx + c \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\frac{\mbox{d}f (z)}{\mbox{d}z}
 & = & \frac{\partial u}{\partial x} + i \, \frac{\partial v}{\partial x} \\
& = & \frac{\partial v}{\partial y} - i \, \frac{\partial u}{\partial y} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\frac{\mbox{d}f (z)}{\mbox{d}z}
 & = & \frac{\partial u}{\partial x} + i \, \frac{\partial v}{\partial x} \\
& = & (2ax + b) (((ax^{2} - ay^{2} + bx + c)^{2} + y^{2} (2ax + b)^{2}) \\
 & & \qquad - \, (ax^{2} - ay^{2} + bx + c) (2 (ax^{2} - ay^{2} + bx + c) + 4ay^{2})) \frac{1}{N^{2}} \\
 & & \qquad \qquad + \, i ((- 2ay) ((ax^{2} - ay^{2} + bx + c)^{2} + y^{2} (2ax + b)^{2}) \\
 & & \qquad \qquad \qquad + \, y (2ax + b)^{2} (2 (ax^{2} - ay^{2} + bx + c) + 4ay^{2})) \frac{1}{N^{2}} \\
& = & (P (Q^{2} + y^{2} P^{2}) - 2 P Q (Q + 2ay^{2}) - 2iay (Q^{2} + y^{2} P^{2}) + 2iy P^{2} (Q + 2ay^{2})) \frac{1}{N^{2}} \\
& = & (P Q^{2} + y^{2} P^{3} - 2 P Q^{2} - 4ay^{2} P Q - 2iay Q^{2} - 2ia y^{3} P^{2} + 2iy P^{2} Q + 4ia y^{3} P^{2}) \frac{1}{N^{2}} \\
& = & (- P Q^{2} + y^{2} P^{3} - 4ay^{2} P Q - 2iay Q^{2} + 2iy P^{2} Q + 2ia y^{3} P^{2}) \frac{1}{N^{2}} \\
& = & - (P Q^{2} - y^{2} P^{3} + 4ay^{2} P Q + 2iay Q^{2} - 2iy P^{2} Q - 2ia y^{3} P^{2}) \frac{1}{N^{2}} \\
& = & - (P (Q^{2} - 2iy P Q - y^{2} P^{2}) + 2iay (Q^{2} - 2iy P Q - y^{2} P^{2})) \frac{1}{N^{2}} \\
& = & - (P + 2iay) (Q^{2} - 2iy P Q - y^{2} P^{2}) \frac{1}{N^{2}} \\
& = & - (P + 2iay) (Q - iy P)^{2} \frac{1}{N^{2}} \\
& = & - (P + 2iay) (Q - iy P)^{2} \frac{1}{((Q + iy P) (Q - iy P))^{2}} \\
& = & - (P + 2iay) (Q - iy P)^{2} \frac{1}{(Q + iy P)^{2} (Q - iy P)^{2}} \\
& = & - \frac{P + 2iay}{(Q + iy P)^{2}} \\
& = & - \frac{2ax + b + 2iay}{(ax^{2} - ay^{2} + bx + c + iy (2ax + b))^{2}} \\
& = & - \frac{2a (x + iy) + b}{(ax^{2} - ay^{2} + bx + c + 2iaxy + iby)^{2}} \\
& = & - \frac{2a (x + iy) + b}{(a (x + iy)^{2} + b (x + iy) + c)^{2}} \\
& = & - \frac{2az + b}{(az^{2} + bz + c)^{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{Noemer}
 & = & 0 \\
az^{2} + bz + c
 & = & 0 \\
z
 & = & \frac{- b \pm \sqrt{b^{2} - 4ac}}{2a} \\
& = & \frac{- b \pm \sqrt{D}}{2a} \\
& = & \frac{- b \pm 0}{2a} \\
& = & - \frac{b}{2a} \\
x + iy
 & = & - \frac{b}{2a} \\
x
 & = & - \frac{b}{2a} \quad \wedge \quad y \, \, \, = \, \, \, 0 \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
p
 & = & - \frac{b}{2a} + i \cdot 0 \\
\end{eqnarray*}
\end{document}
