Question

Prove that 3 + 2√5 is irrational.​

Answer 1

Answer:

Given:3 + 2√5

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

Proof:

Letus 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

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Answer 2

Answer:

To Prove: 3 + 2√5

Step-by-step explanation:

To Prove: 3 + 2√5 is irrational

Proof:

Let  is rational.

A number is said to be rational if it can be expressed in the form p/q where q ≠ 0

Therefore,

We can find two integers p & q where, (q ≠ 0) such that





Since p and q are integers,   will also be rational and therefore,  is rational.

We know that √5 is irrational but according to above statement it has to be rational

So, both the comments are contradictory,

Hence, the number should have been irrational to make the statement correct.

Therefore,  is irrational

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