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  4. The roots of the equation f(x)=a(x-b)(x-c)+b(x-c)(x-a)+c(x-a)(x-b)=0 ( a, b, c are distinct and real ) are always :

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Q. The roots of the equation $f(x)=a(x-b)(x-c)+b(x-c)(x-a)+c(x-a)(x-b)=0$ ( $a, b, c$ are distinct and real ) are always :

Complex Numbers and Quadratic Equations

Solution:

$a\left(x^2-(b+c) x+b c\right)+b\left(x^2-(c+a) x+a c\right)+c\left(x^2-(a+b) x+a b\right)=0 $
$(a+b+c) x^2-2 x(a b+b c+c a)+3 a b c=0$
$D =4(a b+b c+c a)^2-12 a b c(a+b+c) $
$=4\left[a^2 b^2+b^2 c^2+c^2 a^2+2 a b c(a+b+c)-3 a b c(a+b+c)\right] $
$ =4\left[a^2 b^2+b^2 c^2+c^2 a^2-a b c(a+b+c)\right]$
$ =2\left[(a b-b c)^2+(b c-c a)^2+(c a-a b)^2\right]>0$