We are given the equation:

\[
x - 9 = \sqrt{x - 3}
\]

To solve this equation, let's follow these steps:

### Step 1: Square both sides
To eliminate the square root, square both sides of the equation:

\[
(x - 9)^2 = (\sqrt{x - 3})^2
\]

This simplifies to:

\[
(x - 9)^2 = x - 3
\]

### Step 2: Expand the left-hand side
Now, expand \( (x - 9)^2 \):

\[
(x - 9)^2 = x^2 - 18x + 81
\]

So the equation becomes:

\[
x^2 - 18x + 81 = x - 3
\]

### Step 3: Rearrange the equation
Move all terms to one side of the equation:

\[
x^2 - 18x + 81 - x + 3 = 0
\]

Simplify:

\[
x^2 - 19x + 84 = 0
\]

### Step 4: Solve the quadratic equation
Now solve the quadratic equation \(x^2 - 19x + 84 = 0\) using the quadratic formula:

\[
x = \frac{-(-19) \pm \sqrt{(-19)^2 - 4(1)(84)}}{2(1)}
\]

Simplifying the discriminant:

\[
x = \frac{19 \pm \sqrt{361 - 336}}{2}
\]

\[
x = \frac{19 \pm \sqrt{25}}{2}
\]

\[
x = \frac{19 \pm 5}{2}
\]

This gives two possible solutions:

\[
x_1 = \frac{19 + 5}{2} = \frac{24}{2} = 12
\]
\[
x_2 = \frac{19 - 5}{2} = \frac{14}{2} = 7
\]

### Step 5: Check for extraneous solutions
We need to check both solutions in the original equation to make sure they are valid:

- For \( x = 12 \):
  \[
  12 - 9 = \sqrt{12 - 3} \implies 3 = \sqrt{9} \implies 3 = 3
  \]
  This is a valid solution.

- For \( x = 7 \):
  \[
  7 - 9 = \sqrt{7 - 3} \implies -2 = \sqrt{4} \implies -2 = 2
  \]
  This is not valid.

### Final Solution:
The only solution is \( \boxed{x = 12} \).

Do you want more details or have any questions?

Here are 5 related questions:
1. How do we identify extraneous solutions when solving square root equations?
2. Can we solve this type of equation graphically?
3. What is the importance of squaring both sides when solving square root equations?
4. How do you factor quadratic equations without using the quadratic formula?
5. What are the common mistakes when solving equations involving square roots?

**Tip:** Always check your solutions in the original equation when dealing with square roots to avoid extraneous solutions!

This problem walks through the steps to solve the equation x - 9 = sqrt(x - 3), focusing on how to handle square root equations and apply the quadratic formula. Students will learn to check for extraneous solutions, making it ideal for Grades 9-11.

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