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\section{Quadratic Equations}
The quadratic equation $f(x)=x^2+px+q$ has zero, one or two roots.
To see this we add a term to create a complete square:
\begin{equation}
x^2+px+q=0
\end{equation}
\begin{equation}
x^2+px+\frac{1}{4}p^2 = \frac{1}{4}p^2 -q
\end{equation}
Now we can apply the binomial equation:
\begin{equation}
\left( x+\frac{p}{2} \right)^2 = \frac{1}{4}p^2 -q
\end{equation}
We take the square root on both sides of the equation
\begin{equation}
x+\frac{p}{2} = \sqrt{\frac{p^2}{4} -q}
\end{equation}
to get
\begin{equation}
x_{1,2} = -\frac{p}{2} \pm \sqrt{\frac{p^2}{4} -q}
\end{equation}
The expression in the root
\begin{equation}
D = {\frac{p^2}{4} -q}
\end{equation}
is called ``discriminant''. The quadratic equation has two real roots
for $D>0$ and none for $D<0$. If $D=0$,
we have a double root.
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