Question

Prove that 3+2 root 5 is irrational.​

Answer 1

${ \large{ \sf{ \underbrace{\underline{\bigstar \:Given:}}}}}$

3 + 2√5

${ \large{ \sf{ \underbrace{\underline{\bigstar \:To \:prove:}}}}}$

$\textit{3 + 2√5}$ is an irrational number.

${ \large{ \sf{ \underbrace{\underline{\bigstar \:Proof:}}}}}$

Let us assume that $\sf\color{red}3 + 2√5$is a rational number.

${ \large{ \sf{ \underbrace{\underline{\bigstar \:So,}}}}}$

it can be written in the form a/b

$\boxed{\bf{3 + 2√5 = a/b}}$

Here a and b are coprime numbers and b ≠ 0

$\textbf{Solving 3 + 2√5 = a/b we get}$

$\sf\color{magenta}{2√5 = a/b – 3}$

$\sf\color{magenta}{2√5 = (a-3b)/b}$

$\sf\color{magenta}{√5 = (a-3b)/2b}$

$\small\color{orange}{This \: shows \: (a-3b)/2b \: is \: a \:rational \: number.}$

$\small\color{brown}{But \: we \: know \: that \: √5 \: is \: an \:irrational \: number.}$

$\small\color{green}{So, \: it \: contradicts \: our \:assumption.}$

$\small\color{blue}{Our \: assumption \: of \: 3 + 2√5 is \: a \:rational \:number \:is \: incorrect.}$

$\small\color{olive}{3 + 2√5 \:is\: an\: irrational \:number}$

Hence proved

Thankyou :)

Answer 2

sol:-

let 3+2√5 is rational

let 3+2√5 =p/q (q ≠ 0 )

=> 2√5 =p/q -3

=>2√5 =p-3q/ 2q

2,3,p,q are integers

p-3q/2q is rational

√5 is rational which is not possible

this contradiction has arisen due to our wrong assumption that 3+2√5 is rational

we conclude that 3+2√5 is irrational...

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