Let α , β and γ are the roots of the equation 2 x 3 + 9 x 2 - 27 x - 54 = 0 . If α , β , γ are in geometric progression, then the value of α + β + γ =

Step-by-step solution

Let, α = β r and γ = β r α ⋅ β ⋅ γ = β r ⋅ β ⋅ β r = 54 2 = 2 7 ⇒ β 3 = 27 ⇒ β = 3 α + β + γ = β r + β + β r = - 9 2 ⇒ 3 r + 3 + 3 r = - 9 2 ⇒ 1 r + 1 + r = - 3 2 ⇒ 2 + 2 r + 2 r 2 = - 3 r ⇒ 2 r 2 + 5 r + 2 = 0 ⇒ 2 r + 1 r + 2 = 0 ⇒ r = - 1 2 , - 2 α = β r = 3 - 2 = - 3 2 β = 3 γ = β r = 3 - 2 = - 6 α + β + γ = 3 2 + 3 + 6 = 21 2