Show that the roots of the equation: (x-a) (x -b) +(x- b) (x-c)+ (x- c) (x -a) =0 are always real and these cannot be equal unless a=b=c

Explanation

`3 x^{2}-2 x(a+b+c)+(a b+b c+c a)=0`


Let {D} be the discriminant



D=4(a+b+c)^{2}-12(a b+b c+c a)

`=4(a+b+c)^{2}-3(a b+b c+c a) `

`=2[2 a^{2}+2 b^{2}+2 c^{2}-2 a b-2 b c-2 c a`

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