If α , β and γ are the roots of the equation x 3 - 13 x 2 + 15 x + 189 = 0 and one root exceeds the other by 2, then the value of α + β + γ is equal to

Step-by-step solution

Let, the roots be α , β , α + 2 . S 1 = α + β + α + 2 = 2 α + β + 2 = 13 ⇒ 2 α + β = 11 ⇒ β = 11 - 2 α S 2 = α β + β α + 2 + α + 2 α = 1 5 ⇒ β α + α + 2 + α α + 2 = 1 5 ⇒ 11 - 2 α 2 α + 2 + α α + 2 = 1 5 ⇒ 22 α + 22 - 4 α 2 - 4 α + α 2 + 2 α = 1 5 ⇒ 3 α 2 - 20 α - 7 = 0 ⇒ α - 7 3 α + 1 = 0 ⇒