Question

Prove that (√x+1)+(√x-1) is irrational.​

Answer 1

Answer:

Let √x−1+√x+1 be a rational number which can be expressed as p/q, p and q are integers and coprime. q is not equal to 0

squaring on both sides we get x-1+x+1+2 √x2−1

2x+2 √x2−1 =p2q2

2(x+√x2−1)=p2q2

2(x+√x2−1)q2=p2

this mean 2 divides p2 and also divides p.

then let p=2k for any integer k

then 2(x+√x2−1)=(2k)2q2

2(x+√x2−1)=4k2q2

q2=2k2/(x+√x2−1)

so 2 divides q2 and also q

p and q have common factors 2 which contradicts the fact that p and q are co-primes which is due to our wrong assumption. so 

√x−1+√x+1 is irrational