If a,b, and c are not all equal and α and β be the roots of the equation ax2+bx+c=0, then value of 1+α+α21+β+β2 is

We have α+β=−b/a,αβ=c/aNow,    1+α+α21+β+β2         =1+α+β+α2+β2+αβ+α2β+αβ2+α2β2         =1+(α+β)+(α+β)2−2αβ+αβ[1+α+β+αβ]         =1−ba+b2a2−2ca+ca+ca−ba+c2a2         =1a2a2+b2+c2−bc−ca−ab         =12a2(b−c)2+(c−a)2+(a−b)2≥0