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Difference between revisions of "Quadratic equation"

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(Comment: Characteristic 2 case)
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The roots and coefficients of a quadratic equation are related by (cf. [[Viète theorem|Viète theorem]]):
 
The roots and coefficients of a quadratic equation are related by (cf. [[Viète theorem|Viète theorem]]):
 
\begin{equation}
 
\begin{equation}
x_1+x_2=-\frac{b}{2},\quad x_1x_2=\frac{c}{a}.  
+
x_1+x_2=-\frac{b}{a},\quad x_1x_2=\frac{c}{a}.  
 
\end{equation}
 
\end{equation}
 
The expression $b^2-4ac$ is called the [[Discriminant|discriminant]] of the equation. It is easily proved that $b^2-4ac=(x_1-x_2)^2$, in accordance with the fact mentioned above that the equation has a double root if and only if $b^2=4ac$. Formula \eqref{eq:2} holds also if the coefficients belong to a field with characteristic different from $2$.
 
The expression $b^2-4ac$ is called the [[Discriminant|discriminant]] of the equation. It is easily proved that $b^2-4ac=(x_1-x_2)^2$, in accordance with the fact mentioned above that the equation has a double root if and only if $b^2=4ac$. Formula \eqref{eq:2} holds also if the coefficients belong to a field with characteristic different from $2$.
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====References====
 
====References====
 
<table>
 
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Lidl,  H. Niederreiter,  "Finite fields" , Addison-Wesley  (1983); second edition Cambridge University Press (1996) Zbl 0866.11069</TD></TR>
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<TR>
 +
<TD valign="top">[a1]</TD> <TD valign="top">  R. Lidl,  H. Niederreiter,  "Finite fields" , Addison-Wesley  (1983); second edition Cambridge University Press (1996) Zbl 0866.11069</TD></TR>
 
</table>
 
</table>
  
 
[[Category:Field theory and polynomials]]
 
[[Category:Field theory and polynomials]]

Latest revision as of 07:34, 18 December 2014


An algebraic equation of the second degree. The general form of a quadratic equation is \begin{equation}\label{eq:1} ax^2+bx+c=0,\quad a\ne0. \end{equation} In the field of complex numbers a quadratic equation has two solutions, expressed by radicals in the coefficients of the equation: \begin{equation}\label{eq:2} x_{1,2} = \frac{-b \pm\sqrt{b^2-4ac}}{2a}. \end{equation} When $b^2>4ac$ both solutions are real and distinct, when $b^2<4ac$, they are complex (complex-conjugate) numbers, when $b^2=4ac$ the equation has the double root $x_1=x_2=-b/2a$.

For the reduced quadratic equation \begin{equation} x^2+px+q=0 \end{equation} formula \eqref{eq:2} has the form \begin{equation} x_{1,2}=-\frac{p}{2}\pm\sqrt{\frac{p^2}{4}-q}. \end{equation} The roots and coefficients of a quadratic equation are related by (cf. Viète theorem): \begin{equation} x_1+x_2=-\frac{b}{a},\quad x_1x_2=\frac{c}{a}. \end{equation} The expression $b^2-4ac$ is called the discriminant of the equation. It is easily proved that $b^2-4ac=(x_1-x_2)^2$, in accordance with the fact mentioned above that the equation has a double root if and only if $b^2=4ac$. Formula \eqref{eq:2} holds also if the coefficients belong to a field with characteristic different from $2$.

Formula \eqref{eq:2} follows from writing the left-hand side of the equation as $a(x+b/2a)^2+(c-b^2/4a)$ (splitting of the square).

References

[a1] K. Rektorys (ed.) , Applicable mathematics , Iliffe (1969) pp. Sect. 1.20

Comments

Over a field of characteristic 2 (cf. Characteristic of a field), the solution by completing the square is no longer available. Instead, by a change of variable, the equation may be written either as $$ X^2 + c = 0 $$ or in Artin--Schreier form $$ X^2 + X + c = 0 \ . $$

In the first case, the equation has a double root $c^{1/2}$. In the Artin--Schreier case, the map $A:X \mapsto X^2+X$ is two-to-one, since $A(X+1) = A(X)$. If $\alpha$ is a root of the equation, so is $\alpha+1$. See Artin-Schreier theorem.

References

[a1] R. Lidl, H. Niederreiter, "Finite fields" , Addison-Wesley (1983); second edition Cambridge University Press (1996) Zbl 0866.11069
How to Cite This Entry:
Quadratic equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quadratic_equation&oldid=34668
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article