Namespaces
Variants
Actions

Difference between revisions of "Quadratic equation"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (TeX encoding is done)
Line 1: Line 1:
 +
{{TEX|done}}
 +
 
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}
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076050/q0760501.png" /></td> </tr></table>
+
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}
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076050/q0760502.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
x_{1,2} = \frac{-b \pm\sqrt{b^2-4ac}}{2a}.
 
+
\end{equation}
When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076050/q0760503.png" /> both solutions are real and distinct, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076050/q0760504.png" />, they are complex (complex-conjugate) numbers, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076050/q0760505.png" /> the equation has the double root <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076050/q0760506.png" />.
+
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}
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076050/q0760507.png" /></td> </tr></table>
+
x^2+px+q=0
 
+
\end{equation}
formula (*) has the form
+
formula \eqref{eq:2} has the form
 
+
\begin{equation}
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076050/q0760508.png" /></td> </tr></table>
+
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}{2},\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 $2$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076050/q0760509.png" /></td> </tr></table>
+
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).
 
 
 
 
 
 
====Comments====
 
The expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076050/q07605010.png" /> is called the discriminant of the equation. It is easily proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076050/q07605011.png" />, in accordance with the fact mentioned above that the equation has a double root if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076050/q07605012.png" />. See also [[Discriminant|Discriminant]]. Formula (*) holds also if the coefficients belong to a field with characteristic different from 2.
 
 
 
Formula (*) follows from writing the left-hand side of the equation as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076050/q07605013.png" /> (splitting of the square).
 
  
 
====References====
 
====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>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Rektorys (ed.) , ''Applicable mathematics'' , Iliffe  (1969)  pp. Sect. 1.20</TD></TR></table>

Revision as of 08:00, 12 December 2012


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}{2},\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
How to Cite This Entry:
Quadratic equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quadratic_equation&oldid=14167
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article