Question

prove that 3 + 2√5 is irrational.

Answer 1

Hey there !!

To prove :-
3 + 2√5 is irrational

Lets assume that 3 + 2√5 is rational.

Let ,
3 + 2√5 = r , where r is a rational number

Squaring both the sides ,
[3 + 2√5]² =r²

9 + 12√5 + 20 = r²

29 + 12√5 = r²

12√5 = r² - 29

√5 = r² - 29 /12

Here ,
RHS is purely rational. But , LHS [√3 ] is irrational.
This is a contradiction.
Hence , our assumption is wrong.

Therefore ,
3 + 2√5 is irrational