Question

Prove that $3 + 2\sqrt 5$ is irrational.

Answer 1

Let $3+2\sqrt 5$ be rational.
Therefore, we can find two co-prime integers $a, b\ (b ≠ 0)$ such that
$3+2\sqrt 5=\dfrac{a}{b}$

$⇒2\sqrt 5=\dfrac{𝑎}{𝑏}−3$

⇒$$\sqrt 5=\dfrac{1}{2}(\dfrac{𝑎}{𝑏}−3)

Since a and b are integers, $\dfrac{1}{2} (\dfrac{a}{b} −3)$ will also be rational, and therefore, $\sqrt 5$ is rational.
This contradicts the fact that $\sqrt 5$ is irrational. Hence, our assumption that $3+2\sqrt 5$ is rational is false. Therefore, $3+2\sqrt 5$ is irrational.