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. 

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