To solve the quadratic equation $x^2 - 6x = 9$ by completing the square, follow these steps:

Step 1: Move the constant term to the other side.

We want to isolate the $x$-terms on the left-hand side:

$x^2 - 6x = 9$

Subtract 9 from both sides:

$x^2 - 6x - 9 = 0$

Now add 9 to both sides to move it to the right:

$x^2 - 6x = 9$

Step 2: Complete the square.

To complete the square, we take half of the coefficient of $x$, square it, and add it to both sides. The coefficient of $x$ is $-6$, so:

$\left( \frac{-6}{2} \right)^2 = 9$

Add 9 to both sides of the equation:

$x^2 - 6x + 9 = 9 + 9$

Simplifying the right-hand side:

$x^2 - 6x + 9 = 18$

Step 3: Write the left-hand side as a perfect square.

The left-hand side is now a perfect square:

$(x - 3)^2 = 18$

Step 4: Solve for $x$.

Take the square root of both sides:

$x - 3 = \pm \sqrt{18}$

Simplify $\sqrt{18}$:

$\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$

So, the equation becomes:

$x - 3 = \pm 3\sqrt{2}$

Step 5: Solve for $x$.

Add 3 to both sides:

$x = 3 \pm 3\sqrt{2}$

Thus, the two solutions are:

$x = 3 + 3\sqrt{2} \quad \text{or} \quad x = 3 - 3\sqrt{2}$

These are the final simplified solutions.

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Do you want more details or have any questions?

Here are 5 related questions to expand your understanding:

1. How do you solve $x^2 + 4x = 7$ by completing the square?
2. Can you derive the quadratic formula using completing the square?
3. What happens if the discriminant is negative when solving by completing the square?
4. How can you solve $2x^2 - 8x = 5$ using the completing-the-square method?
5. What are the advantages of completing the square over factoring?

Tip: Completing the square is very useful when the quadratic equation doesn't factor easily or when deriving the quadratic formula itself!