
nature of roots
Find the nature of roots for
(xa)(xb) + (xc)(xa) + (xb)(xc) = 0
I started solving the above equation.
This was deduced to
3x2 – 2(a + b+c)x +(ab+bc+ca) =0
from this, the discreminant
4(a + b+c)2 4(3)( ab+bc+ca)
4[a2+b2+c2+2(ab+bc+ca)3(ab+bc+ca)]
4[a2+b2+c2(ab+bc+ca)]
From this I am not able to say what type are the roots.
Kindly guide me.
Thanks

Re: nature of roots
Completing squares we can write the discriminant $\displaystyle D$ in the form
$\displaystyle D=\ldots=(2abc)^2+3(bc)^2$
This implies $\displaystyle D\geq 0$ for all $\displaystyle a,b,c\in\mathbb{R}$ i.e. all the roots are real. Besides, the roots are equal if and only if $\displaystyle (2abc=0)\;\wedge\; (bc=0)$ or equivalently if and only if $\displaystyle a=b=c$ .

Re: nature of roots
Thanks Mr.Fernando Revilla.