properties of 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 $\u2102$ 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
$${({x}_{1}+{x}_{2})}^{3}={x}_{1}^{3}+3{x}_{1}^{2}{x}_{2}+3{x}_{1}{x}_{2}^{2}+{x}_{2}^{3}=({x}_{1}^{3}+{x}_{2}^{3})+3{x}_{1}{x}_{2}({x}_{1}+{x}_{2}),$$ 
we obtain
$${x}_{1}^{3}+{x}_{2}^{3}={({x}_{1}+{x}_{2})}^{3}3{x}_{1}{x}_{2}({x}_{1}+{x}_{2})={4}^{3}3\cdot 9\cdot 4=44.$$ 
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 $5{x}^{2}+11x16=0$ are 1 and $\frac{16}{5}$.
Title  properties of quadratic equation 

Canonical name  PropertiesOfQuadraticEquation 
Date of creation  20150212 9:55:41 
Last modified on  20150212 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 