properties of quadratic equation

The quadratic equation




with rational, real, algebraic ( or complex coefficientsMathworldPlanetmath (a0) has the following properties:

  • It has in two roots (which may be equal), since the complex numbersMathworldPlanetmathPlanetmath form an algebraically closed field containing the coefficients.

  • The sum of the roots is equal to  -ba,  i.e.  -p.

  • The productMathworldPlanetmath of the roots is equal to  ca,  i.e.  q.

Corollary.  If the leading coefficient and the constant are equal, then the roots are inverse numbers of each other.

Without solving the equation, the value of any symmetric polynomialMathworldPlanetmath of the roots can be calculated.

Example.  If one has to x13+x23, when x1 and x2 are the roots of the equation  x2-4x+9=0,  we have  x1+x2=4  and  x1x2=9.  Because


we obtain


Note.  If one wants to write easily a quadratic equation with rational roots, one could take such one that the sum of the coefficients is zero (then one root is always 1).  For instance, the roots of the equation  5x2+11x-16=0  are 1 and -165.

Title properties of quadratic equation
Canonical name PropertiesOfQuadraticEquation
Date of creation 2015-02-12 9:55:41
Last modified on 2015-02-12 9:55:41
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 14
Author pahio (2872)
Entry type Result
Classification msc 12D10
Related topic VietasFormula
Related topic ValuesOfComplexCosine
Related topic IntegralBasisOfQuadraticField