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

Given: Number $2-3\sqrt{5}$

To do: To prove that the given number is an irrational number.

Let us assume that $2-3\sqrt{5} =x$ and$\ x$ is a rational number.

$\therefore \ 2-x=3\sqrt{5} ,\ 2-x$ would also be a rational number.

$\therefore \frac{\ ( 2-x)}{3} =\frac{( 3\surd 5)}{3} =\sqrt{5}$ 

 

If x is a rational number then $2-x$ is also a rational number and $\frac{2-x}{3}$  also should be a rational number.

But here we find that $\therefore \frac{\ ( 2-x)}{3} =5\ $ and $\sqrt{5}$ can never be rational number. 

Our assumption was wrong x to be a rational number.

Thus it has been proved that $2-3\sqrt{5}$ is an irrational number.