Difference between revisions of "Quadratic equation"
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An [[Algebraic equation|algebraic equation]] of the second degree. The general form of a quadratic equation is | An [[Algebraic equation|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: | 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 | + | 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 | For the reduced quadratic equation | ||
| − | + | \begin{equation} | |
| − | + | x^2+px+q=0 | |
| − | + | \end{equation} | |
| − | formula | + | 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|Viète theorem]]): | The roots and coefficients of a quadratic equation are related by (cf. [[Viète theorem|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|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==== | ||
| + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> K. Rektorys (ed.) , ''Applicable mathematics'' , Iliffe (1969) pp. Sect. 1.20</TD></TR></table> | ||
====Comments==== | ====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==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <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> | ||
| + | </table> | ||
| + | |||
| + | [[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 |
Quadratic equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quadratic_equation&oldid=14167