Question

Prove that 6-2root5 is irrational.

Answer 1

Answer:

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Answer 2

Answer:

let us assume that

$6 -2 \sqrt{5}$is rational number.

hence

$6 - 2 \sqrt{5}$

can be written in the form of

$\frac{a}{b}$

where a,b(b≠0) & co-prime.

$6 - 2 \sqrt{5 } = \frac{a}{b} \\ \\ 6 - \frac{a}{b} = 2 \sqrt{5 } \\ \\ \frac{6b - a}{b} = 2 \sqrt{5 } \\ \frac{6b - a}{2b} = \sqrt{5}$

(here 6,a,b,2 are integer )

&therefore

$\frac{6 b - a}{2b} = rational \\ \\ but \: \sqrt{5} \: is \: irrational \\ \\ this \: contradiction\: is \: due \: to \:</p><p> our \: incorrect \: assumption \: that \: 6 - 2 \sqrt{5} is \: rational \\ \\ \\ \\ therfore \: we \: can \: conclude \: bysaying \: 6 - 2 \sqrt{5} is \: irrationa......$

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