quadratic equation in $\mathbb{C}$QuadraticEquationInMathbbC2007-11-01 17:53:462007-11-05 06:55:22Theoremtaking square root algebraically% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\theoremstyle{definition}
\newtheorem*{thmplain}{Theorem}
The quadratic formula
$$x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}$$
for solving the quadratic equation
\begin{align}
ax^2+bx+c = 0
\end{align}
with real coefficients $a$, $b$, $c$ is valid as well for all complex values of these coefficients ($a \neq 0$), when the square root is determined as is presented in the \PMlinkname{parent entry}{TakingSquareRootAlgebraically}.\\
{\em Proof.} Multiplying (1) by $4a$ and adding $b^2$ to both sides gives an \PMlinkname{equivalent}{Equivalent3} equation
$$4a^2x^2+4abx+4ac+b^2 = b^2$$
or
$$(2ax)^2+2\cdot2ax\cdot{b}+b^2 = b^2-4ac$$
or furthermore
$$(2ax+b)^2 = b^2-4ac.$$
Taking square root algebraically yields
$$2ax+b = \pm\sqrt{b^2-4ac},$$
which implies the quadratic formula.\\
\textbf{Note.} A \PMlinkescapetext{similar} quadratic formula is meaningful besides $\mathbb{C}$ also in other fields with characteristic $\neq 2$\, if one can find the needed ``square root'' (this may require a field extension).