Question

Prove that 3 + 2√5 is irrational​

Answer 1

Step-by-step explanation:

Given:3 + 2√5

To prove:3 + 2√5 is an irrational number.

Proof:

Let us assume that 3 + 2√5 is a rational number.

Soit can be written in the form a/b

3 + 2√5 = a/b

Here a and b are coprime numbers and b ≠ 0

Solving3 + 2√5 = a/b we get,

=>2√5 = a/b – 3

=>2√5 = (a-3b)/b

=>√5 = (a-3b)/2b

This shows (a-3b)/2b is a rational number. But we know that But √5 is an irrational number.

so it contradictsour assumption.

Our assumption of3 + 2√5 is a rational number is incorrect.

3 + 2√5 is an irrational number

Hence proved

Answer 2

TO PROVE :-

• 3 + 2√5 is irrational number.

SOLUTION :-

Let us assume that 3 + 2√5 is a " Rational number ". Any number which can be expressed in the form of p/q where p and q are integers and q is not equal to zero [q ≠ 0].

According to our assumption of 3 + 2√5 as a rational number , so it can be written as,

→ p/q = 3 + 2√5. [ q ≠ 0 ]

→ p/q - 3 = 2√5

→ p/q - 3q/q = 2√5

→ (p - 3q)/q = 2√5

→ (p - 3q)/2q = √5

Here (p - 3q)/2q is in the form of p/q and also [ q ≠ 0 ] and hence as we know that √5 is an irrational number. Therefore our assumption of 3 + 2√5 is a " Rational number " is wrong , Therefore 3 + 2√5 is a " Irrational number "

HENCE PROVED ✅