If α,β,γ  are the roots of x3+px2+qx+r=0, then ∑1α2β2=

The given equation is x3+px2+qx+r=0, α,β,γ  are the roots of the given equation ∴   sum of the roots : α+β+γ=−p Sum of the product of roots taken two at a time : αβ+βγ+γα=q Product of the roots : αβγ=−r ∴∑1α2β2=1α2β2+1β2γ2+1α2γ2=α2+β2+γ2α2β2γ2=(α+β+γ)2−2(αβ+βγ+αγ)(αβγ)2 =(−p)2−2(q)(−r)2 =p2−2qr2