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The equation provided is:

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

We will solve this step-by-step:

### Step 1: Isolate the square root term
Move the 3 from the left side of the equation to the right side by subtracting 3 from both sides:

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

### Step 2: Square both sides to eliminate the square root
To eliminate the square root, square both sides of the equation:

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

This simplifies to:

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

### Step 3: Expand the right side
Now, expand \((x - 3)^2\):

\[
x + 3 = x^2 - 6x + 9
\]

### Step 4: Rearrange the equation into a quadratic form
Move all terms to one side of the equation:

\[
x + 3 - (x^2 - 6x + 9) = 0
\]

Simplifying further:

\[
x + 3 - x^2 + 6x - 9 = 0
\]

Combine like terms:

\[
-x^2 + 7x - 6 = 0
\]

### Step 5: Solve the quadratic equation
The equation now is:

\[
x^2 - 7x + 6 = 0
\]

We can solve this quadratic equation by factoring:

\[
(x - 6)(x - 1) = 0
\]

So, the solutions are:

\[
x = 6 \quad \text{or} \quad x = 1
\]

### Step 6: Check for extraneous solutions
We must check both solutions by substituting them back into the original equation:

- For \(x = 6\):
  \[
  \sqrt{6 + 3} + 3 = 6
  \]
  \[
  \sqrt{9} + 3 = 6 \quad \Rightarrow \quad 3 + 3 = 6 \quad \text{(True)}
  \]

- For \(x = 1\):
  \[
  \sqrt{1 + 3} + 3 = 1
  \]
  \[
  \sqrt{4} + 3 = 1 \quad \Rightarrow \quad 2 + 3 = 1 \quad \text{(False)}
  \]

Thus, \(x = 1\) is an extraneous solution. The only valid solution is \(x = 6\).

### Final Answer:
The solution set is \(\{6\}\).

Would you like more details or have any questions?

Here are some related questions you might find helpful:
1. How do you solve other types of radical equations?
2. Why do extraneous solutions occur in equations involving square roots?
3. What other methods can be used to solve quadratic equations besides factoring?
4. How do you graph the function \(\sqrt{x+3} + 3 = x\)?
5. What are some real-world applications of solving radical equations?

**Tip:** Always check for extraneous solutions when solving radical equations, as squaring both sides can introduce solutions that don't satisfy the original equation.

Solve the radical equation. Check all proposed solutions: √(x + 3) + 3 = x

Learn how to solve the radical equation √(x + 3) + 3 = x by isolating the square root term, squaring both sides, expanding, and solving the resulting quadratic equation. Understand why x = 6 is the valid solution and x = 1 is extraneous.

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