Question

If $a, b, c$ are real, then both the roots of the equation \((x-b)(x-c)+(x-c)(x-a)+(x-a)(x-b)=0\) are

Answer 1

\((x-b)(x-c)+(x-c)(x-a)+(x-a)(x-b)=0\)
\(\Rightarrow3x^2-2(a+b+c)x+(bc+ca+ab)=0\)
\(\Delta = 4(a+b+c)^2-12(bc+ca+ab)\)
\(~~~=4[(a+b+c)^2-3(bc+ca+ab)]\)
\(~~~=4[a^2+b^2+c^2-bc-ca-ab]\)
\(\Delta = 2[(a-b)^2+(b-c)^2+(c-a)^2]\)
\(\Rightarrow \Delta \geq 0\)
\( \Rightarrow\) Roots are real.