Prove that 3+2under root 5 is …

According to the question,we have to Prove  that {tex}( 3 + 2 \sqrt { 5 } ){/tex} is irrational.
We will prove this by  using  contradiction method.
Let us assume  that {tex}( 3 + 2 \sqrt { 5 } ){/tex} is rational.
Then,there exists two  co-prime integers a and b (b ≠ 0) such that,

{tex}\frac { a } { b } = 3 + 2 \sqrt { 5 }{/tex} ⇒ {tex}\frac { a } { b } - 3 = 2 \sqrt { 5 }{/tex}

{tex}\Rightarrow{/tex} {tex}\frac { a - 3 b } { b } = 2 \sqrt { 5 }{/tex}

{tex}\Rightarrow \frac { a - 3 b } { 2 b } = \sqrt { 5 }{/tex} ..... (1)
where a and b are integers.
It means L.H.S of (1) is rational but we know that {tex}\sqrt { 5 }{/tex}  is irrational. It is not possible. Therefore, our supposition is wrong. {tex}( 3 + 2 \sqrt { 5 } ){/tex} cannot be rational.
Hence, {tex}( 3 + 2 \sqrt { 5 } ){/tex} is irrational.