Question:
Prove that $2-3 \sqrt{5}$ is an irrational number.
Solution:
Let us assume that $2-3 \sqrt{5}$ is rational .Then, there exist positive co primes $a$ and $b$ such that
$2-3 \sqrt{5}=\frac{a}{b}$
$3 \sqrt{5}=\frac{a}{b}-2$
$3 \sqrt{5}=\frac{\frac{a}{b}-2}{3}$
$\sqrt{5}=\frac{a-2 b}{3 b}$
This contradicts the fact that $\sqrt{5}$ is an irrational number
Hence $2-3 \sqrt{5}$ is irrational
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