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Kenneth John Odle 2023-07-18 09:27:20 -04:00
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complete-the-square.tex
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@ -3,7 +3,7 @@
\usepackage{amsmath} \usepackage{amsmath}
\usepackage{amssymb} \usepackage{amssymb}
\usepackage{graphicx} \usepackage{graphicx}
%\usepackage{kpfonts} \usepackage{kpfonts}
\usepackage{float} \usepackage{float}
\raggedbottom \raggedbottom
@ -62,7 +62,7 @@
\maketitle \maketitle
\begin{abstract} \begin{abstract}
The purpose of this document is to describe how to ``complete the square''---a common method for factoring quadratic equations. The purpose of this document is to describe how to complete the square---a common method for factoring quadratic equations. It also described how the quadratic formula is derived from the standard form for quadratic equations by completing the square.
It also provides several worked equations to serve as examples. It also provides several worked equations to serve as examples.
\end{abstract} \end{abstract}
@ -78,13 +78,13 @@ When we say ``of the form'' what we mean is that:
\item There is at least one term that contains the variable $x^2$. \item There is at least one term that contains the variable $x^2$.
\end{itemize} \end{itemize}
For example, the equation $3x^2+2x=5$ is a quadratic equation because it is written in the form $ax^2+bx+c=0$. For example, the equation $3x^2+2x-5=0$ is a quadratic equation because it is written in the form $ax^2+bx+c=0$.
However, the equation $5=2-3x^2$ is also a quadratic equation because it can be rewritten as $3x^2+3=0$. In this case, the value for $b$ is zero. (This could also be written as $3x^2+0x+3=0$.) However, the equation $5=2-3x^2$ is also a quadratic equation because it can be rewritten as $3x^2+3=0$. In this case, the value for $b$ is zero. (This could also be written as $3x^2+0x+3=0$.)
\paragraph{Perfect Square Quadratic Equations} The simplest quadratic equation to solve is the type where both sides are a perfect square. A \textbf{perfect square} is an integer whose square root is also an integer, such as 9 or 25, because $\sqrt{9}=3$ and $\sqrt{25}=5$. This term can also apply to a term whose coefficient is a perfect square, such as $16x^2$ and $9x^2$ because $\sqrt{16x^2}=4x$ and $\sqrt{9x^2}=3x$. \paragraph{Perfect Square Quadratic Equations} The simplest quadratic equation to solve is the type where both sides are a perfect square. A \textbf{perfect square} is an integer whose square root is also an integer, such as 9 or 25, because $\sqrt{9}=3$ and $\sqrt{25}=5$. This description can also apply to a term whose coefficient is a perfect square, such as $16x^2$ and $9x^2$ because $\sqrt{16x^2}=4x$ and $\sqrt{9x^2}=3x$.
These are easy to solve because you can solve them by taking the square root of both sides: These are easy to solve because you can simply take the square root of both sides:
\begin{align}\label{eq1} \begin{align}\label{eq1}
\begin{split} \begin{split}
@ -128,9 +128,9 @@ x &= -2
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. 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} \section*{The Quadratic Formula}
For equations that are not easily factored, a general solution called ``the quadratic equation'' can be used to solve any quadratic. For equations that are not easily factored, a general solution called ``the quadratic formula'' can be used to solve any quadratic.
For any equation of the form $ax^2+bx+c=0$, the solution (that is, the value for $x$) can be found by using: For any equation of the form $ax^2+bx+c=0$, the solution (that is, the value for $x$) can be found by using:
@ -138,9 +138,9 @@ For any equation of the form $ax^2+bx+c=0$, the solution (that is, the value for
x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}
\end{equation} \end{equation}
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. 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 equation in section ``\nameref{expqe}'' on page~\pageref{expqe}. We will demonstrate how to obtain the quadratic formula in section ``\nameref{expqe}'' on page~\pageref{expqe}.
\section*{Completing the Square} \section*{Completing the Square}
@ -188,7 +188,7 @@ Conventionally, we would write this as $x=-2+\sqrt{3}, -2-\sqrt{3}$.
x^2+4x+1 &= 0 && && &&\text{Original equation}\\ x^2+4x+1 &= 0 && && &&\text{Original equation}\\
& && && &&a=1\text{ so Step 1 is not needed} \\ & && && &&a=1\text{ so Step 1 is not needed} \\
x^2+4x &= -1 && && &&\text{Step 2}\\ x^2+4x &= -1 && && &&\text{Step 2}\\
x^2+4x &= 3 && && &&\text{Step 3}\\ x^2+4x+4 &= 3 && && &&\text{Step 3}\\
(x+2)^2 &= 3 && && && \text{Step 4}\\ (x+2)^2 &= 3 && && && \text{Step 4}\\
\sqrt{(x+2)^2} &= \pm\sqrt{3} && && && \text{Step 5}\\ \sqrt{(x+2)^2} &= \pm\sqrt{3} && && && \text{Step 5}\\
x+2 &= \pm\sqrt{3} && && && \text{Step 5}\\ x+2 &= \pm\sqrt{3} && && && \text{Step 5}\\
@ -244,9 +244,9 @@ For our fourth example, we will look at an equation where neither $a$ is equal t
\subsection*{Example \#6}\label{ex06} \subsection*{Example \#6}\label{ex06}
\section*{An Explanation of the Quadratic Equation}\label{expqe} \section*{An Explanation of the Quadratic Formula}\label{expqe}
Figuring out the quadratic equation by completing the square from the standard form (i.e., $ax^2+bx+c=0$) is not difficult once you are familiar with this problem solving technique. Most people get hung up doing the math on the coefficients. Figuring out the quadratic formula by completing the square from the standard form (i.e., $ax^2+bx+c=0$) is not difficult once you are familiar with this problem solving technique. Most people get hung up doing the math on the coefficients.
\begin{equation}\label{quadsolv} \begin{equation}\label{quadsolv}
\setlength{\jot}{10pt} % Add space between each equation \setlength{\jot}{10pt} % Add space between each equation