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Prove that 3+2 root 5 is …

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Prove that 3+2 root 5 is irrational

Posted by Sahib Ansari 6 years, 3 months ago

CBSE > Class 10 > Mathematics

1 answers

Sia ? 6 years, 3 months ago

We will prove this by contradiction.
Let us suppose that (3+2 {tex}\sqrt { 5 }{/tex}) is rational.
It means that we have co-prime integers a and b such that
{tex}\frac { a } { b } = 3 + 2 \sqrt { 5 } \quad \frac { a } { b } - 3 = 2 \sqrt { 5 }{/tex}

{tex}\Rightarrow \frac{{a - 3b}}{b} = 2{\sqrt 5 \,{ \Rightarrow }}\frac{{a - 3b}}{{2b}} = \sqrt 5 {/tex} ....(1)
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. (3+2 {tex}\sqrt { 5 }{/tex}) cannot be rational.
Hence, (3+2 {tex}\sqrt { 5 }{/tex}) is irrational.

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CBSE > Class 10 > Mathematics