Let us assume (3 -√15 ) is rational

(Assume)

where a & b are integers (b≠0)

⇒

⇒

Now let’s solve the R.H.S. Of the above equation

Let

Squaring we get

In The above equation since 15 divides p² so it must also divide p

so p is a multiple of 15

let p = 15k where k is an integer

Putting in Equation 1 the value of p we get

15q² = 225k²

⇒ q² = 15k²

Since 15 divides q² so it must also divide q

so q is a multiple of 15

But this contradicts our previously assumed data since we had considered p & q has been resolved in their simplest form and they shouldn't have any common factors.

So √15 is irrational and hence

(3 -√15) is also irrational

Hence Proved