properties of quadratic equation

 $ax^{2}\!+\!bx\!+\!c=0$

or

 $x^{2}\!+\!px+\!q\!=0$

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

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

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 $-\frac{16}{5}$.

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