∵α,β,γ are three consecutive terms of a non- constant G.P. ∴β2=αγ So roots of the equation αx2+2βx+γ=0 are
−2β±2√β2−αγ
2α
=
β
α
∵αx2+2βx+γ=0 and x2+x−1=0 have a common root. ∴ this root satisfy the equation x2+x−1=0 β2−αβ−α2=0 ⇒αγ−αβ−α2=0⇒α+β=γ Now, α(β+γ)=αβ+αγ =αβ+β2=(α+β)β=βγ