Question

The roots of the equation $(x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0$ are

• A. positive
• B. negative
• C. real
• D. imaginary

Answer 1

The correct option is C real

Given equation is written as
$3x^2-2x(a+b+c)+(ab+bc+ca)=0$
Discriminant is $4{(a+b+c)}^2-12(ab+bc+ca)$
$=4(a^2+b^2+c^2-ab-bc-ca)$
$=2[{(a-b)}^2+{(b-c)}^2+{(c-a)}^2]$
=+ve
$\therefore$ Roots are real