Question

Given that `sqrt(5)` is an irrational number, prove that `2 + 3sqrt(5)` is an irrational number.

Answer 1

Let `2 + 3sqrt(5)` be a rational number.

∴ `2 + 3sqrt(5) = p/q`

Where p, q are integers and q ≠ 0.

`3sqrt(5) = p/q - 2`

`sqrt(5) = 1/3 (p/q - 2)`

`sqrt(5) = 1/3 ((p - 2q)/q)`

`sqrt(5) = (p - 2q)/(3q)`

Since p and q are integers.

∴ p – 2q is also an integer, 3q is also an integer.

Therefore, `(p - 2q)/(3q)` is a rational number.

This implies `sqrt(5)` is rational, which contradicts the given information that `sqrt(5)` is irrational.

∴ `2 + 3sqrt(5)` must be irrational.

Hence proved.