Question

Prove that $2-3\sqrt{5}$ is an irrational no.

Answer 1

Let $x=2-3\sqrt{5}$ be a rational number.

$3\sqrt{5}=2-x$

$\sqrt{5}=\frac{2-x}{3}$

Since x is rational, 2-x is rational and hence $\frac{2-x}{3}$ is also rational number

$\Rightarrow \sqrt{5}$ is a rational numbers, which is a contradiction.

Hence $2-3\sqrt{5}$ must be an irrational number.