Question

prove that 3 + root 5 is an irrational number​

Answer 1

Let 3 + √5 = x be rational number.

on squaring both side

x2 = 9 + 5 + 3√5

x2 = 14 + 3√5

3√5 = x2 -14

√5 = x2 - 14 / 3

Since , it is assumed that 3+√5 = x is rational

= x2 - 14 is rational .....1

x2 - is rational

and, so x2 - 14/3 is rational

But √2 is irrational .....2

From1 , x2 is rational and

from 2 , x2 is irrational

Therefore , we arrive at a contradiction

So , our assumption that 3 + √5 is a rational number is wrong

Hence , 3 + √5 is irrational .

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