https://file.aiquickdraw.com/mathbot/file/1730455416572u7r6i01o.jpg

To solve the equation:

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

we'll follow these steps:

### Step 1: Square Both Sides
To remove the square root, square both sides of the equation:

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

This simplifies to:

\[
x + 9 = x^2 - 6x + 9
\]

### Step 2: Rearrange Terms
Bring all terms to one side of the equation to set it equal to zero:

\[
0 = x^2 - 7x
\]

### Step 3: Factor the Equation
Now, factor out \( x \):

\[
x(x - 7) = 0
\]

### Step 4: Solve for \( x \)
This gives us two possible solutions:

1. \( x = 0 \)
2. \( x = 7 \)

### Step 5: Verify the Solutions
We need to verify that these solutions satisfy the original equation because squaring both sides can sometimes introduce extraneous solutions.

#### Check \( x = 0 \):
\[
\sqrt{0 + 9} = 0 - 3
\]
\[
3 = -3 \quad \text{(false)}
\]
So, \( x = 0 \) is not a valid solution.

#### Check \( x = 7 \):
\[
\sqrt{7 + 9} = 7 - 3
\]
\[
\sqrt{16} = 4
\]
\[
4 = 4 \quad \text{(true)}
\]
So, \( x = 7 \) is a valid solution.

### Final Answer:
\[
x = 7
\]

Would you like more details or have any questions?

Here are some related questions you might find useful:
1. What are extraneous solutions, and why do they appear in equations with square roots?
2. How can you solve other types of radical equations?
3. What methods are useful when solving quadratic equations?
4. Why is verifying solutions important in radical equations?
5. How would this problem change if there was an additional constant term on the right side?

**Tip:** Always check your solutions in the original equation when dealing with square roots or other even-power operations, as squaring can introduce false solutions.

Learn how to solve the radical equation sqrt(x + 9) = x - 3 using algebraic techniques. This solution includes squaring both sides, simplifying the resulting quadratic equation, and checking for extraneous solutions. Suitable for students in Grades 8-10.

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