Question

2. Prove that 3 +2√5 is irrational.​

Answer 1

Step-by-step explanation:

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Answer 2

$\underline{\underline{\green{\rm Given:-}}}$

• 3+2√5

$\underline{\underline{\purple{\rm Prove:-}}}$

• Prove 3+2√5 an irrational no.

$\underline{\underline{\orange{\rm Solution:-}}}$

Let, us assume to contrary that 3+2√5 is an Rational number.

Now,

$\sf Let,3 + 2 \sqrt{5} = \frac{a}{b}$

where, a and b are coprimes and b ≠ 0

So,

$\sf \to 2 \sqrt{5} = \frac{a}{b} - 3$

$\sf \to \sqrt{5} = \frac{a}{2b} - \frac{3}{2}$

____________________

Since, a and b are integers, Therefore

$\sf \to \frac{a}{2b} - \frac{3}{2} \: is \: a \: rational \: no.$

$\sf \therefore \sqrt{5} \: is \: a \: rational \: no.$

___________________

$\sf But \sqrt{5} \: is \: an \: irrational \: no.$

So, this shows that our assumption is incorrect

$\sf So,\underline{3 + 2 \sqrt{5}} \: is \: an \: irrational \: no.$

HENCE, PROVED