\documentclass[a4paper]{article}
\begin{document}
\begin{eqnarray*}
f (x)
 & = & \sqrt{ax^{2} + bx + c} \\
a
 & < & 0 \\
D
 & = & b^{2} - 4ac \, \, \, > \, \, \, 0 \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
f (x)
 & = & g (h (x)) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
f^{\prime} (x)
 & = & g^{\prime} (h (x)) h^{\prime} (x) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
p
 & = & \frac{- b + \sqrt{b^{2} - 4ac}}{2a} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
q
 & = & \frac{- b - \sqrt{b^{2} - 4ac}}{2a} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & \int \sqrt{ax^{2} + bx + c} \, \mbox{d}x \\
& = & \int \sqrt{a (x - p) (x - q)} \, \mbox{d}x \\
& = & \int \sqrt{a \left ( x - \frac{- b + \sqrt{b^{2} - 4ac}}{2a} \right ) \left ( x - \frac{- b - \sqrt{b^{2} - 4ac}}{2a} \right )} \, \mbox{d}x \\
& = & \int \sqrt{a \left ( x + \frac{b}{2a} - \frac{\sqrt{b^{2} - 4ac}}{2a} \right )
 \left ( x + \frac{b}{2a} + \frac{\sqrt{b^{2} - 4ac}}{2a} \right )} \, \mbox{d}x \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
x
 & = & u - \frac{b}{2a} \\
u
 & = & x + \frac{b}{2a} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & \int \sqrt{a \left ( x + \frac{b}{2a} - \frac{\sqrt{b^{2} - 4ac}}{2a} \right )
 \left ( x + \frac{b}{2a} + \frac{\sqrt{b^{2} - 4ac}}{2a} \right )} \, \mbox{d}x \\
& = & \int \sqrt{a \left ( u - \frac{b}{2a} + \frac{b}{2a} - \frac{\sqrt{b^{2} - 4ac}}{2a} \right )
 \left ( u - \frac{b}{2a} + \frac{b}{2a} + \frac{\sqrt{b^{2} - 4ac}}{2a} \right )} \, \mbox{d} \left ( u - \frac{b}{2a} \right ) \\
& = & \int \sqrt{a \left ( u - \frac{\sqrt{b^{2} - 4ac}}{2a} \right ) \left ( u + \frac{\sqrt{b^{2} - 4ac}}{2a} \right )} \, \mbox{d}u \\
& = & \int \sqrt{a \left ( u^{2} - \left ( \frac{\sqrt{b^{2} - 4ac}}{2a} \right )^{2} \right )} \, \mbox{d}u \\
& = & \sqrt{- a} \int \sqrt{\left ( \frac{\sqrt{b^{2} - 4ac}}{2a} \right )^{2} - u^{2}} \, \mbox{d}u \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & \sqrt{- a} \int \sqrt{\left ( \frac{\sqrt{b^{2} - 4ac}}{2a} \right )^{2} - u^{2}} \, \mbox{d}u \\
& = & \sqrt{- a} \left ( \frac{1}{2} u \sqrt{\left ( \frac{\sqrt{b^{2} - 4ac}}{2a} \right )^{2} - u^{2}}
 + \frac{1}{2} \left ( \frac{\sqrt{b^{2} - 4ac}}{2a} \right )^{2} \arcsin \frac{2 a u}{\sqrt{b^{2} - 4ac}} \right ) + C \\
& = & \frac{1}{2} \sqrt{- a} \left ( u \sqrt{\left ( \frac{\sqrt{b^{2} - 4ac}}{2a} \right )^{2} - u^{2}}
 + \left ( \frac{\sqrt{b^{2} - 4ac}}{2a} \right )^{2} \arcsin \frac{2 a u}{\sqrt{b^{2} - 4ac}} \right ) + C \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & \frac{1}{2} \sqrt{- a} \left ( u \sqrt{\left ( \frac{\sqrt{b^{2} - 4ac}}{2a} \right )^{2} - u^{2}}
 + \left ( \frac{\sqrt{b^{2} - 4ac}}{2a} \right )^{2} \arcsin \frac{2 a u}{\sqrt{b^{2} - 4ac}} \right ) + C \\
& = & \frac{1}{2} \sqrt{- a} \left ( \left ( x + \frac{b}{2a} \right )
 \sqrt{\left ( \frac{\sqrt{b^{2} - 4ac}}{2a} \right )^{2} - \left ( x + \frac{b}{2a} \right )^{2}}
 + \left ( \frac{\sqrt{b^{2} - 4ac}}{2a} \right )^{2} \arcsin \frac{2 a \left ( x + \frac{b}{2a} \right )}{\sqrt{b^{2} - 4ac}} \right ) + C \\
& = & \frac{\sqrt{- a}}{8 a^{2}} \left ( (2ax + b) \sqrt{b^{2} - 4ac - (2a x + b)^{2}}
 + (b^{2} - 4ac) \arcsin \frac{2 a x + b}{\sqrt{b^{2} - 4ac}} \right ) + C \\
& = & \frac{\sqrt{- a}}{8 a^{2}} \left ( (2ax + b) \sqrt{D - (2a x + b)^{2}} + D \arcsin \frac{2 a x + b}{\sqrt{D}} \right ) + C \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
f (x)
 & = & \frac{\mbox{d}(F (x))}{\mbox{d}x} \\
& = & \frac{\mbox{d}}{\mbox{d}x} \left ( \frac{\sqrt{- a}}{8 a^{2}} \left ( (2ax + b) \sqrt{b^{2} - 4ac - (2a x + b)^{2}}
 + (b^{2} - 4ac) \arcsin \frac{2 a x + b}{\sqrt{b^{2} - 4ac}} \right ) + C \right ) \\
& = & \frac{\sqrt{- a}}{8 a^{2}} \Biggl (
 (2a + 0) \sqrt{b^{2} - 4ac - (2a x + b)^{2}} \\
 & & \qquad \qquad
 + \, (2ax + b) \frac{1}{2} \frac{1}{\sqrt{b^{2} - 4ac - (2a x + b)^{2}}} (0 - 2 (2a x + b) (2a)) \\
 & & \qquad \qquad \qquad
 + \, (b^{2} - 4ac) \frac{1}{\sqrt{1 - \left ( \frac{2 a x + b}{\sqrt{b^{2} - 4ac}} \right )^{2}}} \frac{2 a}{\sqrt{b^{2} - 4ac}} \Biggr ) + 0 \\
& = & \frac{\sqrt{- a}}{4 a} \left (
 \sqrt{b^{2} - 4ac - (2a x + b)^{2}} - \frac{(2ax + b)^{2}}{\sqrt{b^{2} - 4ac - (2a x + b)^{2}}}
 + \frac{b^{2} - 4ac}{\sqrt{b^{2} - 4ac - (2 a x + b)^{2}}} \right ) \\
& = & \frac{\sqrt{- a}}{4 a} \left (
 \frac{b^{2} - 4ac - (2a x + b)^{2}}{\sqrt{b^{2} - 4ac - (2a x + b)^{2}}} - \frac{(2ax + b)^{2}}{\sqrt{b^{2} - 4ac - (2a x + b)^{2}}}
 + \frac{b^{2} - 4ac}{\sqrt{b^{2} - 4ac - (2 a x + b)^{2}}} \right ) \\
& = & \frac{\sqrt{- a}}{4 a} \frac{b^{2} - 4ac - (2a x + b)^{2} - (2ax + b)^{2} + b^{2} - 4ac}{\sqrt{b^{2} - 4ac - (2 a x + b)^{2}}} \\
& = & \frac{\sqrt{- a}}{2 a} \frac{b^{2} - 4ac - (2a x + b)^{2}}{\sqrt{b^{2} - 4ac - (2 a x + b)^{2}}} \\
& = & \frac{\sqrt{- a}}{2 a} \sqrt{b^{2} - 4ac - (2 a x + b)^{2}} \\
& = & \frac{\sqrt{- a}}{2 a} \sqrt{b^{2} - 4ac - 4 a^{2} x^{2} - 4 abx - b^{2}} \\
& = & \frac{\sqrt{- a}}{a} \sqrt{- ac - a^{2} x^{2} - abx} \\
& = & \sqrt{a x^{2} + bx + c} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & - \frac{\sqrt{- a}}{8 a^{2}} \left ( (2ax + b) \sqrt{b^{2} - 4ac - (2a x + b)^{2}}
 + (b^{2} - 4ac) \arcsin \frac{2 a x + b}{\sqrt{b^{2} - 4ac}} \right ) + C \\
& = & - \frac{\sqrt{- a}}{8 a^{2}} \left ( (2ax + b) \sqrt{D - (2a x + b)^{2}} + D \arcsin \frac{2 a x + b}{\sqrt{D}} \right ) + C \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & - \frac{\sqrt{- a}}{8 a^{2}} \left ( (2ax + b) \sqrt{b^{2} - 4ac - (2a x + b)^{2}}
 + (b^{2} - 4ac) \arcsin \frac{2 a x + b}{\sqrt{b^{2} - 4ac}} \right ) + C \\
& = & - \frac{\sqrt{- a}}{8 a^{2}} \left ( (2ax + b) \sqrt{D - (2a x + b)^{2}} + D \arcsin \frac{2 a x + b}{\sqrt{D}} \right ) + C \\
\end{eqnarray*}
\end{document}
