Question

prove that 3+2√5 is irrational​

Answer 1

Let us assume that 3 + 2√5 is rational such that we can write in the form of a/b i.e rational number

$\mathtt{\implies \: 3 + 2\sqrt{5} = \frac{a}{b}}$

$\mathtt{\implies \: 2\sqrt{5} = \frac{a}{b} - 3}$

$\mathtt{\implies \: 2\sqrt{5} = \frac{a - 3b}{b}}$

$\mathtt{\implies \: \sqrt{5} = \frac{a - 3b}{2b}}$

We know that √5 is irrational

But in this case √5 is equal to a rational number which is in the form of a/b

This contradiction arose due to our wrong assumption that 3 + 2√5 is rational

∴ 3 + 2√5 is irrational

Answer 2

Step-by-step explanation:

let assume 3+2√5 is rational

in rational number p\q not equal to zero

so we take p\q is a\b

5√5= a\b

√5a=5b

squaring on both sides

(√5a)²=(5b)²

5a²=25b²

so √5is divisible by 5& multiply by 5

:. our assumption is wrong,it is irrational

so,3+2√5is irrational