Question

Show that the roots of the equations $(x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0$ and they cannot be equal unless $a=b=c$.

Answer 1

For the given LHS to be equal to RHS , we have,

each of the terms of the equation on LHS should be zero i.e(x - c)(x - b) =

(x - c)(x - a) = (x - a)(x - b) = 0

then, either x=b or c, x=c or a, x=a or b

thus we have , x = a = b = c which satisfies the given equation