Show that the roots of (x-b)(x-c) +(x-c)(x-a) +(x-a)(x-b) =0 are real, and that they cannot be equal unless a=b=c.
(x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0
Implies that 3x² -2x(a+b+c) +ab+bc+ca=0
Discriminant= (D) = 4(a+b+c)² - 4*3*(ab+bc+ca)
=4 [a²+b²+c²+2ab+2bc+2ca -3ab-3bc-3ca ]
=4 [ a²+b²+c² -ab-bc-ca ]
=2 [ (a-b)² +(b-c)² +(c-a)² ]
which is always greater then zero so the roots are real .
Roots are equal if D =0
ie (a-b)² +(b-c)² +(c-a)²=0
Since sum of three perfect square is equal to zero so each of them separately equal to zero
So a-b=0 , b-c=0 c-a=0
a=b b=c c=a
So a=b=c
Implies that 3x² -2x(a+b+c) +ab+bc+ca=0
Discriminant= (D) = 4(a+b+c)² - 4*3*(ab+bc+ca)
=4 [a²+b²+c²+2ab+2bc+2ca -3ab-3bc-3ca ]
=4 [ a²+b²+c² -ab-bc-ca ]
=2 [ (a-b)² +(b-c)² +(c-a)² ]
which is always greater then zero so the roots are real .
Roots are equal if D =0
ie (a-b)² +(b-c)² +(c-a)²=0
Since sum of three perfect square is equal to zero so each of them separately equal to zero
So a-b=0 , b-c=0 c-a=0
a=b b=c c=a
So a=b=c