@ -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: