Question

Prove that $6 - 4\sqrt{5}$ is an irrational number, given that $\sqrt{5}$ is an irrational number.

Answer 1

Let $x = 6 - 4\sqrt{5}$.

Assume for contradiction that $x$ is rational, which means $6 - 4\sqrt{5}$ is a rational number.

Rearranging:

$$4\sqrt{5} = 6 - x$$

$$\sqrt{5} = \frac{6 - x}{4}$$

Since $x$ is assumed to be rational, the right-hand side is rational, which implies that $\sqrt{5}$ must be rational.

But $\sqrt{5}$ is irrational, which is a contradiction. Therefore, $6 - 4\sqrt{5}$ must be irrational.