Question

show that 5-2root3 is an irrational number

Answer 1

Let 5-2√3 be a rational number.
A rational number can be written in the form of p/q where p,q are integers and q ≠ 0

5-2√3 = p/q
2√3 = 5 - p/q
2√3 = (5q-p)/q
√3 = (5q-p)/2q

p,q are integers then (5q - p)/2q is a rational number.
Then √3 must be a rational number.
But this contradicts the fact that √3 is an irrational number.
So, our supposition is false.
Hence, 5 - 2√3 is an irrational number.