If α and β are roots of the equation x² + x - 1 = 0, then the equation whose roots are α/β and β/α is:

Question: If α and β are roots of the equation x² + x - 1 = 0, then the equation whose roots are α/β and β/α is:

Solution:
As α and β are roots of x² + x - 1 = 0,
then
⇒ α + β = - ( + 1) = - 1
⇒ αβ = - 1

Now, if (α/β) and (β/α) are roots then,
⇒ Sum of roots = (α/β) + (β/α)
= (α² + β²)/αβ
= [(α + β)² - 2αβ]/αβ
= (- 1)² - 2(-1)]/(-1)
= (1+ 2)/(- 1)
= - 3

Product of roots = (α/β) × (β/α) = 1
Now, then the equation is,
⇒ x² - (Sum of roots)x + Product of roots = 0
⇒ x² - (- 3)x + (1) = 0
⇒ x² + 3x + 1 = 0