Prove that 3+2√5 is irrational.

Let us assume 3+2√5 is rational. So we can write this number as 3+2√5 = a/b — (1) Here a and b are two co-prime numbers and b is not equal to zero. Simplify the equation (1), subtract 3 from both sides, we get 2√5 = a/b - 3 2√5 = (a - 3b)/b Now divide by 2 we get √5 = (a - 3b)/(2b) Here a and b are integers so (a - 3b)/(2b) is a rational number, so √5 should be a rational number. But √5 is an irrational number, so it is a contradiction. Therefore, 3+2√5 is an irrational number.