Question

The sum of two roots of a quadratic equation is 5 and sum of their cubes is 35, find the equation.

Answer 1

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.