If x² - 3x + 1 = 0, and x > 1, then what is the value of x - (1/x)?

Question: If x² - 3x + 1 = 0, and x > 1, then what is the value of x - (1/x)?

Solution:
We are given:
x² - 3x + 1 = 0
⇒ x - 3 + 1/x = 0
∴ x + 1/x = 3

Now,
(a - b)² = (a + b)² - 4ab
⇒ (x - 1/x)² = (x + 1/x)² - 4 × x × (1/x) [Here, a = x, b = 1/x]

Substitute the values:
(x - 1/x)² = 3² - 4
⇒ (x - 1/x)² = 9 - 4
⇒ (x - 1/x)² = 5
∴ x - 1/x = √5