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

Question

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, 23, 17, 13, 19

MARKS App · Accepted Answer

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 ⇒ α = 7 or - 1 3 . α = 7 , β = 11 - 2 α = 11 - 14 = - 3 , γ = α + 2 = 9 α = - 1 3 , β = 11 - 2 α = 11 + 2 3 = 35 3 , γ = α + 2 = 5 3 . Since, α β γ = - 189 , hence we will take the first case. α + β + γ = 7 + - 3 + 9 = 1 9

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