Given α and β are the roots of the quadratic equation x2−4x+k=0(k≠0). If αβ,αβ2+α2β,α3+β3 are in geometric progression, then the value of 7 k equals______.

Given αβ;αβ(α+β);α3+β3are in G.P.α+β=4; αβ=k;αβ2+α2β=αβ(α+β)=4kα3+β3=(α+β)3−3αβ(α+β)=64−3k(4)=4(16−3k)∴ k;4k;4(16−3k) are in G.P. 16k2=4k(16−3k)4k(4k−16+3k)=0k=0;k=167