Question

prove that 3+2root5 irrational number

Answer 1

let 3+2√5 be rational number.
so, 3+2√5= p/q (p,q are integers and q≠0)
=>2√5=p/q -3
=>2√5=(p-3q)/q
=>√5=(p-3q)/2q
since √5 is irrational and (p-3q)/2q is rational.
=>√5≠(p-3q)/2q
so, our assumption is wrong.
so, 3+2√5 is irrational number.

Answer 2

Let 3+2√5 be rational.
3+2√5 = a/b ,where a and b are co-prime and (b≠0)
→2√5=a/b – 3
→2√5=a–3b/b
→√5=a–3b/2b
Here, a–3b/2b is rational but,√5 is irrational.
Since, rational≠ irrational
This is a contradiction because of our incorrect consumption.
So, we conclude that 3+2√5 is irrational.