Question

The roots of the equation x² + 6x – 4 = 0 are α, β. Find the quadratic equation whose roots are α²β and β²α

Answer 1

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

α²β + β²α = αβ(α + β)

= –  4(– 6) = 24

α²β × β²α = α³β³ = (αβ)³ = (– 4)³ = – 64

∴ The required equation = x² – 24x – 64 – 0