Question

prove that 3 +2 root 5 is irrational .( proceed with justifying root 5 as irrational)

Answer 1

let us assume that 3+2√5 is a rational number.
3+2√5=a/b
2√5=a/b-3
square the right side to get another integer c
2√5=a×a/b×b-9
2√5=a²/b²-9 hence that right side is a rational number.
but our assumption that 3+2√5 is a rational number is false
so it is a irrational number

Answer 2

Step-by-step explanation:

3 + 2√5 is irrational number

__________ [TO PROVE]

Let us assume that, 3 + 2√5 is a rational number

=> 3 + 2√5 = \dfrac{a}{b}

b

a

Here, a and b are co-prime numbers.

=> 2√5 = \dfrac{a}{b}

b

a

- 3

=> 2√5 = \dfrac{a\:-\:3b}{b}

b

a−3b

=> √5 = \dfrac{a\:-\:3b}{2b}

2b

a−3b

Here..

\dfrac{a\:-\:3b}{2b}

2b

a−3b

is rational number.

So, √5 is also a rational number.

But we know that √5 is irrational number.

So, our assumption is wrong.

3 + 2√5 is irrational number.

___________ [HENCE PROVED]