2+√5

(v) 2+√5  

Sol. If one of the root of the quadratic equation is 2+√5  ,

then the other root is 2 - √5  

Let α= 2+√5  and β= 2 - √5  

∴ α + β  = 2+√5  + 2 - √5  = 4

and  α β  =  (2+√5  ) × (2 - √5  )

= (2)²  –  (√5)²

= 4 – 5

= - 1

We know that, Quadratic equation is given by,

x² – (Sum of the roots)x + Product of the roots = 0

∴ x² –( α  + β)x + αβ = 0

∴ x² – (4)x + (-1) = 0

∴  x² -  4x  - 1  = 0

∴  The required quadratic equation is  x² -  4x  - 1  = 0