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\title{Completing the Square}
\author{Kenneth John Odle}
\date{16 July 2023}
\begin{document}
\maketitle
\begin{abstract}
The purpose of this document is to describe how to ``complete the square''---a common method for factoring quadratic equations.
It also provides several worked equations to serve as examples.
\end{abstract}
\section{Quadratic Equations}
\textbf{Quadratic equations} are equations of the form $ax^2+bx+c=0$.
The simplest quadratic equation to solve is the type where both sides are a perfect square, because you can solve them by taking the square root of both sides:
\paragraph{Note:} Because we are taking the square root of a constant, we must include both the positive and negative values of the square root as the solution, hence $\pm3$. Substituting either $3$ or $-3$ into $x$ in the original equation results in a value of 36. (In story problems, the situation may mean that we can safely ignore one of these values.)
Using either of these values for $x$ in equation \ref{eq2} will result in one of the factors being equal to zero, meaning both sides of the equation will be zero.
\section{The Quadratic Equation}
For equations that are not easily factored, a general solution called ``the quadratic equation'' can be used to solve any quadratic.
In reality, the quadratic equation is a generalized form of the solving technique called ``completing the square''. Using the quadratic equation 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.
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.
\paragraph{Note:} Our goal is to make the \textit{left} side of the equation into a square. However, the right side will often not be a perfect square, as in equation set~(\ref{eq1}) where the right side was $36$, which is simply $6^2$. This means that we will have a $\sqrt{~~}$ sign in our solution.
The left side is now a perfect square, even though it doesn't look like it. Because $x^2+4x+4=(x+2)^2$ we can rewrite it as a perfect square (step four): \[(x+2)^2=3\]