Solution:

Finding the roots of x³ 9x² - x - 9 = 0 can be easily done by factorizing the equation. However, we will use the method of solving cubic equations in this case.

Step 1: Check for rational roots
- We can use the rational root theorem to find any rational roots of the equation.
- The possible rational roots of the equation are ±{1, 3, 9} / ±{1}.
- By testing these values, we can see that none of them are roots of the equation.

Step 2: Use the cubic formula
- The cubic formula is used to find the roots of a cubic equation of the form ax³ + bx² + cx + d = 0.
- The roots of this equation can be found using the following formula:

x = [(-b ± √(b² - 4ac - 3b³ / a)) / (2a)]^(1/3) - [a / (3b)]

- In this equation, a = 1, b = 9, c = -1, and d = -9.
- Substituting these values into the cubic formula, we get:

x = [(-9 ± √(9² - 4(1)(-1) - 3(9)³ / 1)) / (2(1))]^(1/3) - [1 / (3(9))]

- Simplifying this equation, we get:

x = [-3 ± √33]^(1/3) - 1/27

Step 3: Simplify the roots
- We can simplify the roots by using the fact that:

(√a ± √b)³ = a√a ± 3a√b + 3b√a ± b√b

- Using this formula, we can simplify the roots to:

x₁ = -1 + (√33 + 3√3) / 6
x₂ = -1 - (√33 - 3√3) / 6
x₃ = -1 + (√33 - 3√3) / 6

Step 4: Check the roots
- We can check the roots by substituting them back into the original equation.
- By doing this, we can see that all three roots satisfy the equation.

Therefore, the roots of x³ 9x² - x - 9 = 0 are:

x₁ = -1 + (√33 + 3√3) / 6
x₂ = -1 - (√33 - 3√3) / 6
x₃ = -1 + (√33 - 3√3) / 6.