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Kenneth John Odle 11 months ago
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complete-the-square.tex

#### 38 complete-the-square.tex View File

 @ -7,6 +7,8 @@ \usepackage{float} \raggedbottom   \usepackage{enumitem}   % Where are our images? \graphicspath{{images/}}   @ -63,9 +65,20 @@ It also provides several worked equations to serve as examples.   \section{Quadratic Equations}   \textbf{Quadratic equations} are equations of the form $ax^2+bx+c=0$.  \textbf{Quadratic equations} are equations of the form $ax^2+bx+c=0$, or that can be written in that form.    When we say of the form'' what we mean is    \begin{itemize}[noitemsep]  \item The letters $a, ~b,$ and $c$ represent constants, and that both $b$ and $c$ can be zero.  \item There is one term that contains the variable $x^2$ \end{itemize}    For example, the equation $3x^2+2x=5$ is a quadratic equation because it is written in the form $ax^2+bx+c=0$.    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 $3x^2+0x+3=0$.)   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{Perfect Square Quadratic Equations} 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:   \begin{align}\label{eq1} \begin{split} @ -80,7 +93,7 @@ x &= \pm3   \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.)   \paragraph{Factorable equations} The other type of quadratic equation is one that can easily be solved by factoring. For example, \paragraph{Factorable Quadratic Equations} The other type of quadratic equation is one that can easily be solved by factoring. For example,   \begin{align}\label{eq2} \begin{split} @ -156,7 +169,7 @@ and then solve for $x$ (step six): $x=\pm\sqrt{3} -2$   Conventionally, we would write this as $x=-2+\sqrt{3}, -2-\sqrt{3}$.   \krule{6pt}{6pt} \newpage   \noindent{}Here is the entire sequence all together:   @ -176,18 +189,27 @@ x &= -2\pm\sqrt{3} && && &&\text{Step 6}   \hrule   \section{Example \#2} \subsection{Example \#2}   For our second example, we will look at an equation where $a$ is not equal to 1.   \section{Example \#3} \subsection{Example \#3}   For our third example, we will look at an equation where the right side is not a perfect square.   \section{Example \#4} \subsection{Example \#4}   For our fourth example, we will look at an equation where neither $a$ is equal to 1 nor the right side is a perfect square.   \pagestyle{lastpage}% Remove the header from the last page; comment this out if the document ends on an odd-numbered page \section{More Worked Examples}   \subsection{Example \#5.1}   \subsection{Example \#5.2}   \section{An Explanation of the Quadratic Equation}   %\pagestyle{lastpage} % Remove the header from the last page; comment this out if the document ends on an odd-numbered page   \end{document}