Question

The sum of two roots of a quadratic equation is 7, and the sum of their cubes is 91; find the equation.

Answer 1

Let α and β be the roots of the quadratic equation.

From the given information

α + β = 7       ...(1)

α³ + β³ = 91     ...(2)

(α + β)³ = α³ + β³ + 3αβ(α + β)

∴ α³ + β³ = (α + β)³ − 3αβ(α + β)

91 = (7)³ − 3αβ(7)         ...[From (1)]

91 = 343 − 21αβ

∴ 343 − 21αβ = 91        ...[From (2)]

∴ 21αβ = 343 − 91

∴ 21αβ = 252

∴ αβ = `252/21`

∴ αβ = 12         ...(3)

The required quadratic equation is

x² − (α + β)x + αβ = 0

∴ x² − 7x + 12 = 0      ...[From (1) and (3)]

∴ The required equation is x² − 7x + 12 = 0.