# MathJax v3 beta: MathML input, HTML output test

When $a\ne 0$, there are two solutions to $a{x}^{2}+bx+c=0$ and they are $x=\frac{-b±\sqrt{{b}^{2}-4ac}}{2a}\text{.}$

## The Lorenz Equations

$\begin{array}{rl}\stackrel{˙}{x}& =\sigma \left(y-x\right)\\ \stackrel{˙}{y}& =\rho x-y-xz\\ \stackrel{˙}{z}& =-\beta z+xy\end{array}$

## The Cauchy-Schwarz Inequality

${\left(\sum _{k=1}^{n}{a}_{k}{b}_{k}\right)}^{\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}2}\le \left(\sum _{k=1}^{n}{a}_{k}^{2}\right)\left(\sum _{k=1}^{n}{b}_{k}^{2}\right)$

## A Cross Product Formula

${\mathbf{V}}_{1}×{\mathbf{V}}_{2}=|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ \frac{\mathrm{\partial }X}{\mathrm{\partial }u}& \frac{\mathrm{\partial }Y}{\mathrm{\partial }u}& 0\\ \frac{\mathrm{\partial }X}{\mathrm{\partial }v}& \frac{\mathrm{\partial }Y}{\mathrm{\partial }v}& 0\end{array}|$

## The probability of getting $k$ heads when flipping $n$ coins is:

$P\left(E\right)=\left(\genfrac{}{}{0}{}{n}{k}\right){p}^{k}\left(1-p{\right)}^{n-k}$

## An Identity of Ramanujan

$\frac{1}{\left(\sqrt{\varphi \sqrt{5}}-\varphi \right){e}^{\frac{2}{5}\pi }}=1+\frac{{e}^{-2\pi }}{1+\frac{{e}^{-4\pi }}{1+\frac{{e}^{-6\pi }}{1+\frac{{e}^{-8\pi }}{1+\dots }}}}$

## Maxwell's Equations

$\begin{array}{rl}\mathrm{\nabla }×\stackrel{\to }{\mathbf{B}}-\phantom{\rule{thinmathspace}{0ex}}\frac{1}{c}\phantom{\rule{thinmathspace}{0ex}}\frac{\mathrm{\partial }\stackrel{\to }{\mathbf{E}}}{\mathrm{\partial }t}& =\frac{4\pi }{c}\stackrel{\to }{\mathbf{j}}\\ \mathrm{\nabla }\cdot \stackrel{\to }{\mathbf{E}}& =4\pi \rho \\ \mathrm{\nabla }×\stackrel{\to }{\mathbf{E}}\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}\frac{1}{c}\phantom{\rule{thinmathspace}{0ex}}\frac{\mathrm{\partial }\stackrel{\to }{\mathbf{B}}}{\mathrm{\partial }t}& =\stackrel{\to }{\mathbf{0}}\\ \mathrm{\nabla }\cdot \stackrel{\to }{\mathbf{B}}& =0\end{array}$

## In-line Mathematics

Finally, while display equations look good for a page of samples, the ability to mix math and text in a paragraph is also important. This expression $\sqrt{3x-1}+\left(1+x{\right)}^{2}$ is an example of an inline equation. As you see, MathJax equations can be used this way as well, without unduly disturbing the spacing between lines.