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'quadratic equation in $\mathbb{C}$'
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| Title of object: |
quadratic equation in $\mathbb{C}$ |
| Canonical Name: |
QuadraticEquationInMathbbC |
| Type: |
Theorem |
| Created on: |
2007-11-01 17:53:46 |
| Modified on: |
2007-11-03 14:48:01 |
| Classification: |
msc:12D99, msc:30-00 |
| Synonyms: |
quadratic equation in $\mathbb{C}$=quadratic equation |
Revision comment (for changes between this and next version):
Preamble:
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\usepackage{amssymb}
\usepackage{amsmath}
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%\usepackage{psfrag}
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%\usepackage{graphicx}
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\usepackage{amsthm}
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%\usepackage{xypic}
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% define commands here
\theoremstyle{definition}
\newtheorem*{thmplain}{Theorem}
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Content:
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).
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