Question

pove that 3+2√5 is irrational ​

Answer 1

To Prove: 3+2√5 is irrational

Let 3+2√5 is a rational number.

We know that we can write a rational number in the form of p/q. Where q ≠ 0.

So,

3+2√5 = p/q ( Where q ≠ 0 )

2√5 = p/q - 3

Taking LCM

2√5 = ( p - 3q )/3

After solving it we will get

√5 = $\boxed{\bf{\frac{p - 3q}{6}}}$

But √5 is an irrational number and we cannot write it in the form of p/q.

Hence, it contradicts the fact that √5 is irrational.

So , 3+2√5 is irrational .

Answer 2

To prove :- 3 + 2√5 is an irrational

Solution :-

Let us considered that 3 + 2√5 is an rational number.

Therefore, 3 + 2√5 can be written as p/q form.

Where q is not equal to 0

Therefore,

3 + 2√5 = p/q

2√5 = p - 3q / q

2√5 = p- 3q/q

√5 = 1/2 × p - 3q/ q

√5 = p - 3q / 2q

√5 is an irrational number and p - 3q / 2q is a rational number.

√5 cannot be equal to p - 3q / 2q.

Therefore, We can conclude that our assumption was incorrect.

Hence proved that 3 + 2√5 is an irrational number.