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# Prove That $$3 + 2\sqrt{5}$$ is Irrational.

Let us suppose that $$3 + 2\sqrt{5}$$ is rational
So there should be two co- prime integers (p and q, where q $$\ne$$ 0 ) such that:

$$3 + 2\sqrt{5} = {{p}\over{q}}$$
$$\Rightarrow 2\sqrt{5} ={{p}\over{q}} - 3$$
$$\Rightarrow 2\sqrt{5} ={{p}\over{q}} - 3$$
$$\Rightarrow \sqrt{5} =\frac{1}{2}(\frac{p}{q} - 3)$$ .................(i)

p and q are integers which means $$\frac{1}{2}(\frac{p}{q} - 3)$$ is a rational number.

But we know $$\sqrt{5}$$ is irrational.

This is not possible since LHS $$\ne$$ RHS.
That proves that our assumption that $$3 + 2\sqrt{5}$$ is rational is wrong.

Hence it is proved that $$3 + 2\sqrt{5}$$ is irrational.