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Question
The sum of two roots of a quadratic equation is 5 and sum of their cubes is 35, find the equation.
Solution
Let α and β be the roots of the quadratic equation.
From the given information
α + β = 5 ...(1)
α3 + β3 = 35 ...(2)
(α + β)3 = α3 + β3 + 3αβ(α + β)
∴ α3 + β3 = (α + β)3 − 3αβ(α + β)
35 = (5)3 − 3αβ(5) ...[From (1)]
35 = 125 − 15αβ
∴ 125 − 15αβ = 35 ...[From (2)]
∴ 15αβ = 125 − 35
∴ 15αβ = 90
∴ αβ = 6 ...(Dividing by 15) ...(3)
The required quadratic equation is
x2 − (α + β)x + αβ = 0
∴ x2 − 5x + 6 = 0 ...[From (1) and (3)]
∴ The required equation is x2 − 5x + 6 = 0.
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