Prove that 3+√5 is a irrational

let us suppose 3+root 5 is rational.
=>3+root 5 is in the form of p/q where p and q are integers and q is not =0
=>root5=p/q-3
​=>root 5=p-3q/q
as p, q and 3 are integers p-3q/3 is a rational number.
=>root 5 is a rational number.
but we know that root 5 is an irrational number.
this is an contradiction.
this contradiction has arisen because of our wrong assumption that 3+root 5 is a rational number.
hence 3+ root 5 is an irrational number.