Prove that 3+2?5 is irrational

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.

So it can be written in the form a/b
3 + 2√5 = a/b
Here a and b are coprime numbers and b ≠ 0
Solving 3 + 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 of 3 + 2√5 is a rational number is incorrect.
3 + 2√5 is an irrational number

Hence proved

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