Question

prove that 3+2√5 is irrational?​

Answer 1

Prove that 3+2√5 is irrational.

Take that 3 + 2√5 a rational number.

$➝ \: 3 + 2 \sqrt{5} = \frac{a}{b}$

Where a & b are two co - prime numbers and (b≠0).

$➝ \: 3 + 2 \sqrt{5} = \frac{a}{b} \\ ➝ \: 2 \sqrt{5} = \frac{a}{b} - 3$

$➝ \: 2 \sqrt{5} = \frac{a - 3b}{b} \\ ➝ \: \sqrt{5} = \frac{a - 3b}{2b}$

• Here a-3b/2b is rational.
• But √5 is irrational.

Since Rational ≠ Irrational

This is a contradiction.

Therefore, our assumption is incorrect.

3 + 2√5 is irrational.

Hence Proved.