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Kenneth John Odle 2023-07-18 10:55:15 -04:00
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complete-the-square.tex
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\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage{kpfonts}
%\usepackage{kpfonts}
\usepackage{float}
\raggedbottom
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\title{Completing the Square}
\author{Kenneth John Odle}
\date{17 July 2023}
\date{18 July 2023}
\begin{document}
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\section{Quadratic Equations}
\textbf{Quadratic equations} are equations of the form $ax^2+bx+c=0$, or that can be written in that form.
\textbf{Quadratic equations} are equations of the form $ax^2+bx+c=0$, or equations that can be written in that form.
When we say ``of the form'' what we mean is that:
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In reality, the quadratic formula is the result of solving the general form quadratic equation $ax^2+bx+c=0$ by completing the square. Using the quadratic formula is generally much easier (it can be programmed into some calculators and spreadsheets, for instance), but completing the square is used in certain calculus problems and for graphing some functions.
We will demonstrate how to obtain the quadratic formula in section ``\nameref{expqe}'' on page~\pageref{expqe}.
We will demonstrate how to obtain the quadratic formula in the section titled ``\nameref{expqe}'' on page~\pageref{expqe}.
\section*{Completing the Square}
The general procedure for completing the square is to first make the left side of the equation into a perfect square (the right side will just be a number), and then solving it as we did in \ref{eq1}. This is not a difficult process, but most students tend to get stuck on the first step---making the left side into a perfect square.
The general procedure for completing the square is to first make the left side of the equation into a perfect square (the right side will just be a number), and then solving it as we did in equation set (\ref{eq1}) on page~\pageref{eq1}. This is not a difficult process, but most students tend to get stuck on the first step---making the left side into a perfect square.\\
Here is the general procedure:
\noindent{}Here is the general procedure:
\begin{enumerate}
\item If the value for $a$ is \textit{not} 1, divide both sides of the equation by $a$. (We want the $x^2$ term to be by itself.)