Prove that the following number is irrational

Foundation

Mathematics Foundation

Properties of Real Number

Irrational Number

Rationalisation

Question

Prove that the following number is irrational $(2 + \sqrt{5})$

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$(2 + \sqrt{5})$
Let us assume that $(2 + \sqrt{5}) (2 + \sqrt{5})$
is rational
Subtract given numbers from 2, considering 2 is rational number as we know,difference of two rational numbers is rational.
$(2 + \sqrt{5}) (2 + \sqrt{5})$ is rational
but $\sqrt{5}$ is irrational
Which is contradictory to our assumption.
So $(2 + \sqrt{5})$ is an irrational.