Question

Prove that 3+2V5 is irrational.

NEET

Maths

Solution

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Let us assume on the contrary that ( 3+2 sqrt{5} ) is rational. Then there exist co-prime positive integers ( a ) and ( b ) such that ( 3+2 sqrt{5}=frac{a}{b} ) ( Rightarrow quad 2 sqrt{5}=frac{a}{b}-3 quad Rightarrow quad sqrt{5}=frac{a-3 b}{2 b} ) ( Rightarrow sqrt{5} ) is rational ( therefore a, b ) are integers ( therefore frac{a-3 b}{2 b} ) is a rational This contradicts the fact that ( sqrt{5} ) is irrational. So, our supposition is incorrect. Hence, ( 3+2 sqrt{5} ) is an irrational number.