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%% Copyright (C) 2019 by Walter Daems <walter.daems@uantwerpen.org>
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\documentclass[a4paper,10pt]{article}

\usepackage{a4wide}
\usepackage[english]{babel}
\usepackage[external]{exsol}

\title{External example, from the \textsf{ExSol} package}
\author{Walter Daems}
\setlength{\parindent}{0em}
\begin{document}
\maketitle

\section{Introduction}

In this text we explain how to solve second-order polynomial
equations.

\section{Solving second-order polynomial equations}

\begin{informulacollectiononly}
\section*{Solving second-order polynomial equations}
\end{informulacollectiononly}
\begin{informulacollection}
  The roots of the following equation
  \begin{equation}
    a x^2 + bx + c = 0
  \end{equation}
  can be determined as:
  \begin{equation}
    x_{1,2} = \frac{-b \pm \sqrt{b^2 - 4 a c}}{2 a}
  \end{equation}
\end{informulacollection}

\begin{exercises}[columns = 2]

  \begin{exercise}
    Solve the following equation for $x \in C$, with $C$ the set of
    complex numbers:
    \begin{equation}
      5 x^2 -3 x = 5
    \end{equation}
  \end{exercise}
  \begin{solution}
    Let's start by rearranging the equation, a bit:
    \begin{eqnarray}
      5.7 x^2 - 3.1 x &=& 5.3\\
      5.7 x^2 - 3.1 x -5.3 &=& 0
    \end{eqnarray}
    The equation is now in the standard form:
    \begin{equation}
      a x^2 + b x + c = 0
    \end{equation}
    For quadratic equations in the standard form, we know that two
    solutions exist:
    \begin{equation}
      x_{1,2} = \frac{ -b \pm \sqrt{d}}{2a}
    \end{equation}
    with
    \begin{equation}
      d = b^2 - 4 a c
    \end{equation}
    If we apply this to our case, we obtain:
    \begin{equation}
      d = (-3.1)^2 - 4 \cdot 5.7 \cdot (-5.3) = 130.45
    \end{equation}
    and
    \begin{eqnarray}
      x_1 &=& \frac{3.1 + \sqrt{130.45}}{11.4} = 1.27\\
      x_2 &=& \frac{3.1 - \sqrt{130.45}}{11.4} = -0.73
    \end{eqnarray}
    The proposed values $x = x_1, x_2$ are solutions to the given equation.
  \end{solution}
  \begin{exercise}
    Consider a 2-dimensional vector space equipped with a Euclidean
    distance function. Given a right-angled triangle, with the sides
    $A$ and $B$ adjacent to the right angle having lengths, $3$ and
    $4$, calculate the length of the hypotenuse, labeled $C$.
  \end{exercise}
  \begin{solution}
    This calls for application of Pythagoras' theorem, which
    tells us:
    \begin{equation}
      \left\|A\right\|^2 + \left\|B\right\|^2 = \left\|C\right\|^2
    \end{equation}
    and therefore:
    \begin{eqnarray}
      \left\|C\right\|
      &=& \sqrt{\left\|A\right\|^2 + \left\|B\right\|^2}\\
      &=& \sqrt{3^2 + 4^2}\\
      &=& \sqrt{25} = 5
    \end{eqnarray}
    Therefore, the length of the hypotenuse equals $5$.
  \end{solution}
\end{exercises}
And now, we can come to conclusion.

\section{Conclusion}
Solving second-order polynomial equations is very easy.
\end{document}
\endinput
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