Question

Both roots of the equation (x - a) (x - b) + (x - b) (x - c) + (x - c)(x - a) = 0 are always

• A. positive
• B. negative
• C. real
• D. none of these

Answer 1

The correct option is C real

$(x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0$
$3x^2-2(a+b+c)x+(ab+bc+ca)=0$
$D=4(a+b+c{)}^2-12(ab+bc+ca)$
$=4[a^2+b^2+c^2-ab-bc-ca]$
$D=2\left[\right(a-b{)}^2+(b-c{)}^2+(c-a{)}^2]$
$D\ge 0$
Root must be real.