If the roots are given, the quadratic equation is x2 – (sum of the roots) x + product the roots = 0.
For the given equation.
x2 + 6x – 4 = 0
α + β = -6
αβ = -4
(i) α2 + β2 = (α + β)2 – 2αβ
= (-6)2 – 2(-4) = 36 + 8 = 44
α2β2 = (αβ)2 = (-4)2 = 16
∴ The required equation is x2 – 44x – 16 = 0.
(ii) 2/α and 2/β
(iii) α2β and β2α
α2 β + β2 α = αβ(α + β)
= -4(-6) = 24
α2 β x β2 α = α3 β3 = (αβ)3 = (-4)3 = -64
∴ The required equation = x2 – 24x – 64 – 0.
