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Question
Sum of the roots of the quadratic equation is 5 and sum of their cubes is 35, then find the quadratic equation
Solution
Let α and β be the roots of the quadratic equation.
According to the given conditions,
α + β = 5 and α3 + β3 = 35
Now, (α + β)3 = α3 + 3α2β + 3αβ2 + β3
∴ (α + β)3 = α3 + β3 + 3αβ(α + β)
∴ (5)3 = 35 + 3αβ(5)
∴ 125 = 35 + 15αβ
∴ 125 – 35 = 15αβ
∴ 15αβ = 90
∴ αβ = `90/15`
∴ αβ = 6
∴ The required quadratic equation is
x2 − (α + β)x + αβ = 0
∴ x2 − 5x + 6 = 0
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