\begin{document}
 \maketitle
 \tableofcontents
 
 \section{Introduction} %The section name will appear
             % in the table of contents.
 \label{sec:Introduction} %Labels are useful in
             % cross-referencing.
 Hello, world!  This is a short example further illustrating the power of \LaTeX.  \LaTeX\ is especially useful to writers who care about mathematics but it is useful to a much broader range of writers than just those.  A second section appears in Section~\ref{sec:FurtherDevelopments} on page~\pageref{sec:FurtherDevelopments}.  Note that the use of the package \textbf{hyperref} lets us follow these cross references very easily in programs such as \emph{yap} and Adobe \emph{Acrobat Reader}.
 
 
 \begin{figure}
  \begin{center}
   \includegraphics[width=3in]{Quadratic}
  \end{center}
  \caption{A Quadratic Polynomial:  finding the roots}
  \label{fig:Quadratic}
 \end{figure}

 
 To illustrate the mathematical power of \LaTeX, here is a short derivation of the quad\-ratic formula, usually taught in high school algebra.  Figure~\ref{fig:Quadratic} shows a particular quadratic polynomial of the type $y=ax^2+b+c$ where $a=3$, $b=2$, and $c=-4$.  A problem of frequent interest is to find all the points where the polynomial is zero.  The graph reveals two such points.  One near $x=-1.5$ and one near $x=+0.8$.  The quadratic equation derived here lets us compute the values more exactly.
  \begin{align} % Starts an aligned, numbered equation.
  0 &= a x^2 + b x + c \label{eq:QuadraticPolynomial}
 \end{align}
 
 We start by factoring $a$ out of the polynomial and completing the square.
 \begin{align}
   0 &= a \left( x^2 + \frac{b}{a} x \right)+ c \notag \\
   &= a \left( x^2 + \frac{b}{a} x  + \left( \frac{b}{2a}
     \right)^2 \right)+ c - a \left( \frac{b}{2a} \right)^2 \notag \\
   a \left( x + \frac{b}{2a} \right)^2  &=
    \frac{a b^2 - 4 a^2 c}{4 a^2}. \notag
 \end{align}
 
 A little further manipulation yields
 \begin{align}
   \left( x + \frac{b}{2a} \right)^2  &=
    \frac{b^2 - 4 a c}{4 a^2}. \notag
 \end{align}
 
 Taking the square root on both sides of the equation gives us
 \begin{align}
   x + \frac{b}{2a} &= \pm \sqrt{ \frac{b^2 - 4 a c}{4 a^2} } \notag \\
   x &= -\frac{b}{2a} \pm \frac{1}{2a} \sqrt{ b^2 - 4 a c }. \notag
 \end{align}
 
 The right-hand side can then be put over a common denominator, yielding
 \begin{align}
   x &= \frac{-b \pm \sqrt{ b^2 - 4 a c } }{2a} . \label{eq:QuadraticEquation}
 \end{align} % Ends an aligned, numbered equation.
 
 Applying this to the case illustrated in Figure~\ref{fig:Quadratic}, we have
  \begin{align}
   x &= \frac{-2 \pm \sqrt{ 2^2 - 4 (3) (-4) } }{2(3)} \notag \\
    &= \frac{-2 \pm \sqrt{ 52} }{6} \notag \\
    &= -\frac{1}{3} \pm \frac{\sqrt{13}}{3}  \notag \\
    &= -1.53518 \textrm{ or } 0.86852 .
 \end{align} % Ends an aligned, numbered equation.

 
 In equation~\eqref{eq:QuadraticPolynomial} we see a very general quadratic polynomial, set equal to zero.  The derivation seeks all points $x$ which satisfy this equation.  Equation~\eqref{eq:QuadraticEquation} shows the result in a form familiar to students of high-school algebra.
 
 It's quite easy to include descriptions of \LaTeX\ in a document, too.  For example, the \verb+\label+ commands in the file allow cross-references to each equation, shown here using the \textbf{amsmath} macro \verb+\eqref{}+.  The \textbf{hyperref} package makes it easy to follow them in programs such as \emph{yap} and Adobe \emph{Acrobat Reader}.
 
\section{Further Developments}
\label{sec:FurtherDevelopments}
This section is present only in order to show how to add additional sections.  It is easy to make a cross-reference to another section.  For example, we can refer to Section~\ref{sec:Introduction} on page~\pageref{sec:Introduction}.
\end{document}