# Three famous mathematical formulas

Prerequisites: Before attempting this assessment you should have already worked through all the articles in this module, and also have an understanding of HTML basics (study Introduction to HTML). To have a play with some MathML and test your new-found knowledge.

## A small math article

The goal is to rewrite the following math article using HTML and MathML: Although you don't need to be familiar with LaTeX, it might be useful to know the LaTeX source from which it was generated:

\documentclass{article}

\usepackage{amsmath}
\usepackage{amssymb}

\begin{document}

To solve the cubic equation $t^3 + pt + q = 0$ (where the real numbers
$p, q$ satisfy ${4p^3 + 27q^2} > 0$) one can use Cardano's formula:

$\sqrt[{3}]{ -\frac{q}{2} +\sqrt{\frac{q^2}{4} + {\frac{p^{3}}{27}}} }+ \sqrt[{3}]{ -\frac{q}{2} -\sqrt{\frac{q^2}{4} + {\frac{p^{3}}{27}}} }$

For any $u_1, \dots, u_n \in \mathbb{C}$ and
$v_1, \dots, v_n \in \mathbb{C}$, the Cauchy–Bunyakovsky–Schwarz
inequality can be written as follows:

$\left| \sum_{k=1}^n {u_k \bar{v_k}} \right|^2 \leq { \left( \sum_{k=1}^n {|u_k|} \right)^2 \left( \sum_{k=1}^n {|v_k|} \right)^2 }$

Finally, the determinant of a Vandermonde matrix can be calculated
using the following expression:

$\begin{vmatrix} 1 & x_1 & x_1^2 & \dots & x_1^{n-1} \\ 1 & x_2 & x_2^2 & \dots & x_2^{n-1} \\ 1 & x_3 & x_3^2 & \dots & x_3^{n-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & x_n & x_n^2 & \dots & x_n^{n-1} \\ \end{vmatrix} = {\prod_{1 \leq {i,j} \leq n} {(x_i - x_j)}}$

\end{document}


## Starting point

To get this assessment started, you can rely on our usual HTML template. By default it uses UTF-8 encoding, special Web fonts for the <body> and [itex] tags (with similar look & feel as the LaTeX output). The goal is to replace the question marks ??? with actual MathML content.

<!DOCTYPE html>
<html lang="en-US">
<meta charset="utf-8" />
<title>Three famous mathematical formulas</title>
rel="stylesheet"
href="https://fred-wang.github.io/MathFonts/LatinModern/mathfonts.css" />
<body class="htmlmathparagraph">
<p>
To solve the cubic equation ??? (where the real numbers ??? satisfy ???)
one can use Cardano's formula: ???
</p>

<p>
For any ??? and ???, the Cauchy–Bunyakovsky–Schwarz inequality can be
written as follows: ???
</p>

<p>
Finally, the determinant of a Vandermonde matrix can be calculated using
the following expression: ???
</p>
</body>
</html>


## Hints and tips

• Start by inserting empty [itex] tags, deciding whether they should have a display="block" attribute or not.
• Check the text used and find their Unicode characters ("−", "ℂ", "∑", ...).
• Analyze the semantics of each portion of text (variable? operator? number?) and determine the proper token element to use for each of them.
• Look for advanced constructions (fractions? roots? scripts? matrices?) and determine the proper MathML element to use for each of them.
• Don't forget to rely on <mrow> for grouping subexpressions.
• Pay attention to stretchy and large operators!
• Use the W3C validator to catch unintended mistakes in your HTML/MathML markup.
• If you are stuck, or realize how painful it is to write MathML by hand, feel free to use tools to help write MathML such as TeXZilla.