Form Quadratic Equation Whose Roots Are

Form the quadratic equation whose roots are: (i) √3 and 3√3 (ii) 2 + √5 and 2 – √5

Answers:

(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

x² – (α + β)x +  αβ = 0

⇒ x² – 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

x²  – (α + β)x  +  αβ = 0

⇒  x²  – 4x – 1 = 0