If x² - √5x + 1 = 0, and x > 1, then what is the value of x - (1/x)?

Question: If x² - √5x + 1 = 0, and x > 1, then what is the value of x - (1/x)?

Solution:
Given,
x² - √5x + 1 = 0
⇒ x - √5 + 1/x = 0 [dividing both side by x]
⇒ x + (1/x) = √5

Now,
{x - (1/x)}² = {x + (1/x)}² - 4 . x . (1/x) [∵ (a - b)² = (a + b)² - 4ab]
⇒ {x - (1/x)}² = (√5)² - 4
⇒ {x - (1/x)}² = 5 - 4
⇒ {x - (1/x)}² = 1
∴ x - (1/x) = 1 [∵ x > 1, so taking the positive value]