Question

prove that 3+2√5 is irrational​

Answer 1

Answer :

Let 3+2√5 be a rational number equal to $\dfrac{p}{q}$

And p and q be co-primes i.e HCF(p,q) = 1 and q ≠0

=> $\dfrac{p}{q} = 3 + 2 \sqrt{5}$

=> $\dfrac{p}{q} - 3 = 2 \sqrt{5}$

=> $\dfrac{p - 3q}{q} = 2 \sqrt{5}$

=> $\dfrac{p - 3q}{2q} = \sqrt{5}$

Here, $\dfrac{p - 3q}{2q}$ is rational.

But the fact is √5 is irrational.

=> Rational ≠ Irrational

Therefore, our assumption is wrong.

Hence, 3+2√5 is irrational