To solve the quadratic equation $x^2 - 6x + 6 = 0$ by completing the square, follow these steps:

1. Start with the given equation: $x^2 - 6x + 6 = 0$

2. Move the constant to the other side: $x^2 - 6x = -6$

3. To complete the square, we need to add a specific number to both sides. The number to add is based on the coefficient of $x$ (which is $-6$).

   Take half of the coefficient of $x$, which is $\frac{-6}{2} = -3$, and then square it: $(-3)^2 = 9$

4. Add 9 to both sides of the equation to complete the square: $x^2 - 6x + 9 = -6 + 9$ Simplifying the right-hand side: $x^2 - 6x + 9 = 3$

So, the number that must be added to complete the square is 9.

Would you like further details on how to finish solving the equation?

Here are 5 related questions:

1. How do you factor the completed square form $x^2 - 6x + 9$?
2. Can completing the square method be used for any quadratic equation?
3. What are other methods to solve quadratic equations besides completing the square?
4. How does completing the square relate to the quadratic formula?
5. What are real-world applications of solving quadratic equations?

Tip: When completing the square, always remember to add the same value to both sides of the equation to maintain equality.