Question

Prove that 3 + 2​ \sqrt{5} is irrational​

Answer 1

Proof :

Without losing our generosity, we consider that 3 + 2√5 is rational. Then the number can be expressed in the form

3 + 2√5 = a/b

where a, b are integers with non-zero b

or, 2√5 = a/b - 3

or, 2√5 = (a - 3b)/b

or, √5 = (a - 3b)/2b

Since a, b are integers, (a - 3b) and 2b are also intgers, then (a - 3b)/2b is rational which leads to a contradiction to the fact that √5 is irrational.

Thus, our assumption is wrong.

Hence, 3 + 2√5 is irrational. (proved)