\documentclass[a4paper,11pt]{article} \begin{document} \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. \end{document}