Let α , β , γ be the real roots of the equation, x 3 + a x 2 + b x + c = 0 , ( a , b , c ∈ R and a , b ≠ 0 ). If the system of equations (in, u , v , w ) given by α u + β v + γ w = 0 , β u + γ v + α w

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Let α , β , γ be the real roots of the equation, x 3 + a x 2 + b x + c = 0 , ( a , b , c ∈ R and a , b ≠ 0 ). If the system of equations (in, u , v , w ) given by α u + β v + γ w = 0 , β u + γ v + α w = 0 , γ u + α v + β w = 0 has non-trivial solution, then the value of a 2 b is

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Equation x 3 + a x 2 + b x + c = 0 has roots α , β , γ . Therefore, α + β + γ = - a α β + β γ + γ α = b Since the given system of equations has non-trivial solutions, we have α β γ β γ α γ α β = 0 or α 3 + β 3 + γ 3 - 3 α β γ = 0 or α + β + γ α 2 + β 2 + γ 2 - α β - β γ - γ α = 0 or α + β + γ α + β + γ 2 - 3 α β + β γ + γ α = 0 ⇒ - a a 2 - 3 b = 0 or a 2 / b = 3

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