Question

The roots of the equation x² + 6x – 4 = 0 are α, β. Find the quadratic equation whose roots are `2/alpha` and `2/beta`

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

`2/alpha + 2/beta = (2beta + 2alpha)/(alphabeta)`

= `(2(alpha+beta))/(alphabeta)`

= `(2(- 6))/(- 4)`

= `(- 12)/(- 4)`

= 3

`2/alpha + 2/beta = 4/(alphabeta)`

= `4/(- 4)`

= – 1

∴ The required equation is x² – 3x – 1 = 0.