If the roots of the equation x 3 + b x 2 + c x + d = 0 are in arithmetic progression, then b , c and d satisfy the relation

Question

If the roots of the equation x 3 + b x 2 + c x + d = 0 are in arithmetic progression, then b , c and d satisfy the relation, 2 b 2 - 27 d = 9 b c, 2 b 3 - 27 d = 9 b c, 2 b 2 + 27 d = 9 b c, 2 b 3 + 27 d = 9 b c

MARKS App · Accepted Answer

Let α , β , γ be the required roots α + β + γ = - b , α β + β γ + γ α = c , α β γ = - d Also, α + γ = 2 β ⇒ 3 β = - b β ( α + γ ) + γ α = c ⇒ - b 3 2 - b 3 + γ α = c ⇒ 2 b 2 9 - d β = c ⇒ 2 b 2 9 - d - b 3 = c ⇒ 2 b 2 9 + 3 d b = c ⇒ 2 b 3 + 27 d = 9 b c

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