prove that 3 + 2√5 is irrational.
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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
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
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