Question

Prove that 3 + 2`sqrt5` is irrational

Answer 1

if possible let a = 3 + 2`sqrt5` be a rational number.

Squaring a² = (3+2`sqrt5`)²

a² = 29 +12`sqrt5`

`sqrt5` = `(a^2 - 29)/12` ........(1)

Since a is a rational number the expression `(a^2-29)/12` is also rational number.

⇒ `sqrt5` is a rational number.

This is a contradiction. Hence, 3 + 2`sqrt5` is irrational.

Hence Proved.