Question

If α and β are zeroes of a quadratic polynomial px² + qx + 1. Form a quadratic polynomial whose zeroes are `2/α` and `2/β`.

Answer 1

Given: Let α and β be the zeros of px² + qx + 1. Then `α + β = -q/p` and `αβ = 1/p`.

Step-wise calculation:

1. Sum of the new zeros:

`2/α + 2/β = (2(α + β))/(αβ)`

= `(2(-q/p))/(1/p)`

= –2q

2. Product of the new zeros:

`(2/α)(2/β) = 4/(αβ)`

= `4/(1/p)`

= 4p

3. Form the monic quadratic x² – (sum)x + (product): 

x² – (–2q)x + 4p 

= x² + 2qx + 4p

A quadratic polynomial whose zeros are `2/α` and `2/β` is x² + 2qx + 4p more generally k(x² + 2qx + 4p) for any nonzero constant k.