x² + 1/x² = 18. Then what is the value of x - 1/x?

Let's first simplify the given equation x² + 1/x² = 18.

Multiplying through by x², we get:
x^4 + 1 = 18x²

Rearranging terms, we get:
x^4 - 18x² + 1 = 0

Let y = x², then the equation becomes:
y² - 18y + 1 = 0

Solving this quadratic equation for y using the quadratic formula:
y = (18 ± √(18² - 4*1*1)) / 2
y = (18 ± √(324 - 4)) / 2
y = (18 ± √320) / 2
y = (18 ± 4√5) / 2

y = 9 ± 2√5

Since y = x², we have two possible values for x:
x = √(9 + 2√5) or x = √(9 - 2√5)

Now, let's find the value of x - 1/x for each case:

For x = √(9 + 2√5):
x - 1/x = √(9 + 2√5) - 1/√(9 + 2√5)
x - 1/x = √(9 + 2√5) - √(9 - 2√5)
x - 1/x = √(9 + 2√5) - √(9 - 2√5)

For x = √(9 - 2√5):
x - 1/x = √(9 - 2√5) - 1/√(9 - 2√5)
x - 1/x = √(9 - 2√5) - √(9 + 2√5)
x - 1/x = √(9 - 2√5) - √(9 + 2√5)

Therefore, the value of x - 1/x can be either √(9 + 2√5) - √(9 - 2√5) or √(9 - 2√5) - √(9 + 2√5), depending on the value of x.

x² - 1/x² = 18
1 = 18 (not true)

x - 1/x has no value