Given, α,β and 2β are the real roots of the equation x3−9x2+k=0 where k∈R−{0}. Then, α+β+2β=9⇒α+3β=9......(i) and and α⋅β⋅2β=−k ⇒2αβ2=−k .....(ii) Product of two roots αβ+β⋅2β+2β⋅α=0 ⇒3αβ+2β2=0⇒β(3α+2β)=0 β≠0 because k≠0 so, 3α+2β=0.....(iii) From Eqs. (i) and (iii), 7β=27⇒14β=54