properties of quadratic equation

The quadratic equation

$$a{x}^2+bx+c=0$$

or

$$x^2+px+q=0$$

with rational, real, algebraic (http://planetmath.org/AlgebraicNumber) or complex coefficients ($a\ne 0$) has the following properties:

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

• The sum of the roots is equal to  $-\frac{b}{a}$,  i.e.  $-p$.

• The product of the roots is equal to  $\frac{c}{a}$,  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 polynomial of the roots can be calculated.

Example.  If one has to $x_1^3+x_2^3$, when $x_1$ and $x_2$ are the roots of the equation  $x^2-4x+9=0$,  we have  $x_1+x_2=4$  and  $x_1{x}_2=9$.  Because

$${(x_1+x_2)}^3=x_1^3+3x_1^2{x}_2+3x_1{x}_2^2+x_2^3=(x_1^3+x_2^3)+3x_1{x}_2(x_1+x_2),$$

we obtain

$$x_1^3+x_2^3={(x_1+x_2)}^3-3x_1{x}_2(x_1+x_2)=4^3-3\cdot 9\cdot 4=-44.$$

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  $5x^2+11x-16=0$  are 1 and $-\frac{16}{5}$.

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