Question

prove that 3 + 2 root 5 is an irrational number
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Answer 1

Let's assume to reach the contradiction that 3+2√5 is rational.

Let x = 3+2√5, where x is a rational number which values the RHS.

$\displaystyle x=3+2\sqrt{5} \\ \\ \\ x-3=2\sqrt{5} \\ \\ \\ \frac{x-3}{2}=\sqrt{5}$

At the last line, it seems that √5 can be written in p/q form (as a fraction), where p = x - 3  and q = 2.

But √5 can never be written in p/q form because it is an irrational number.

As a contradiction occurs at the final step, we can conclude the answer with the fact that 3+2√5 won't be a rational number.

Hence proved!!!

Answer 2

clearly sum of rational and irrational number is irrational
3 is rational
2√5 is irrational number
3 +2√5 is irrational