Question

prove that 3+2 root 5 is irrational
pls answer fast with full details and step by step

Answer 1

Step-by-step explanation:

it is a contridiction method

let us assume that 3+2root 5 is rational no

so , 3+2root5 =a/b

2root5=a/b

2 root 5=a/b-3/1

2 root5=a-3b/b

root5=a-3b/2b

so,root 5 ia irrational and

a-3b/2b is a rational number

so assumption is wrong

3+2root 5 is a irrational no

Answer 2

Let us assume, to the contrary, that 3+2√5 is rational.

So we can express this number in terms of a/b (a and b are coprime, a, b are integers, b Is not equal to 0)

So 3+2√5 = a/b

2√5 = a+3a/b

√5 = a+3a/2b

In the RHS, a, b, 3, and 2 are all integers. So a+3a/2b is a rational number.

Since we equate RHS to LHS, √5 is rational. (1)

Let us assume so. Then √5=p/q (p and q are prime, p and q are integers, q is not equal to 0)

So √5q=p

Squaring on both sides,

5q^2 = p^2

q^2 can be any number, but since we multiply it by 5, 5q^2 is divisible by 5.

Since LHS = RHS, p^2 is also divisible by 5.

So p is divisible by 5.

Let us now assume p to be a number 5r, where r is some other integer.

Squaring on both sides, p^2 = 25r^2

5q^2 = 25r^2

q^2 = 5r^2

r^2 can be any number, but since we multiply it by 5, 5r^2 is divisible by 5.

Since LHS = RHS, q^2 is also divisible by 5.

So p and q have two common factors 1 and 5.

Unfortunately, this contradicts the fact that p and q are co primes.

This contradiction has arisen due to our incorrect assumption that √5 is rational.

So √5 is irrational.

Unfortunately, we have made another contradiction. At (1), we have stated that √5 is rational, whereas, in fact, it is irrational.

This contradiction has arisen due to our incorrect assumption that 3+2√5 is rational.

So 3+2√5 is irrational.

Hence Proved.