Prove that $5 - 2\sqrt{3}$ is an irrational number. It is given that $\sqrt{3}$ is an irrational number.

Question

Accepted Answer

Step 1: Assuming $5 - 2\sqrt{3}$ to be a rational number.
Let $5 - 2\sqrt{3} = \frac{a}{b}$ where $a$ and $b$ are integers and $b 
eq 0$
Step 2: $\Rightarrow \sqrt{3} = \frac{5b - a}{2b}$
Here RHS is rational but LHS is irrational.
Step 3: Therefore our assumption is wrong.
Hence, $5 - 2\sqrt{3}$ is an irrational number.

Question

Solution