(i) Let α, β be the roots of the required quadratic equation :
Then, α = √3 and β = 3√3
α + β = √3 + 3√3 and αβ = √3 × 3√3
∴ α + β = 4√3 and αβ = 9
Required quadratic equation
x^{2} – (α + β)x + αβ = 0
⇒ x^{2} - 4√3x + 9 = 0
(ii) Let α, β be the given roots.
Then α = 2 + √5 and β = 2 - √5
α + β = 2 + √5 + 2 - √5 = 4
and αβ = (2 + √5)(2 - √5)
⇒ α + β = 4 and αβ = (2)^{2} - (√5)^{2}
⇒ α + β = 4 and αβ = 4 - 5
⇒ α + β = 4 and αβ = - 1
Required quadratic equation
x^{2} - (α + β)x + αβ = 0
⇒ x^{2} – 4x – 1 = 0