Find all real solutions of the equation. (Enter your answers as a comma-separated list. If there is no real solution, enter NO REAL SOLUTION.)
 

(9x + 49)^1/2
 + 7 = x

We are tasked with finding the real solutions of the equation:

\[
\sqrt{9x + 49} + 7 = x
\]

### Steps to solve:

1. **Isolate the square root:**

Subtract 7 from both sides of the equation:

\[
\sqrt{9x + 49} = x - 7
\]

2. **Square both sides:**

To eliminate the square root, square both sides of the equation:

\[
(\sqrt{9x + 49})^2 = (x - 7)^2
\]

This simplifies to:

\[
9x + 49 = (x - 7)^2
\]

3. **Expand the square on the right-hand side:**

\[
9x + 49 = x^2 - 14x + 49
\]

4. **Simplify the equation:**

Subtract \(9x + 49\) from both sides to set the equation equal to zero:

\[
0 = x^2 - 14x + 49 - 9x - 49
\]

Simplify further:

\[
0 = x^2 - 23x
\]

5. **Factor the quadratic equation:**

\[
x(x - 23) = 0
\]

6. **Solve for \(x\):**

The solutions to this equation are:

\[
x = 0 \quad \text{or} \quad x = 23
\]

### 7. **Check for extraneous solutions:**

Since we squared both sides in the process, we must check both potential solutions by substituting them back into the original equation.

- For \(x = 0\):

\[
\sqrt{9(0) + 49} + 7 = 0 \quad \Rightarrow \quad \sqrt{49} + 7 = 0 \quad \Rightarrow \quad 7 + 7 = 0 \quad (\text{False})
\]

Thus, \(x = 0\) is not a valid solution.

- For \(x = 23\):

\[
\sqrt{9(23) + 49} + 7 = 23 \quad \Rightarrow \quad \sqrt{207 + 49} + 7 = 23 \quad \Rightarrow \quad \sqrt{256} + 7 = 23 \quad \Rightarrow \quad 16 + 7 = 23
\]

This is true, so \(x = 23\) is a valid solution.

### Final Answer:

The real solution is:
\[
x = 23
\]

---

Would you like further clarification on any of the steps?

### Related Questions:
1. What happens if there is more than one square root in an equation?
2. How do you handle extraneous solutions in different types of equations?
3. What is the effect of squaring both sides of an equation on its solutions?
4. Can you explain how to factor quadratic equations in more detail?
5. How do you verify solutions for equations involving square roots?

### Tip:
Always check for extraneous solutions when solving equations involving square roots, as squaring both sides can introduce false solutions.

Find all real solutions of the equation. (Enter your answers as a comma-separated list. If there is no real solution, enter NO REAL SOLUTION.) (9x + 49)^1/2 + 7 = x

Learn how to solve the equation (9x + 49)^1/2 + 7 = x with a detailed, step-by-step approach. This problem covers algebraic concepts, including square roots and quadratic equations, and shows how to handle extraneous solutions. Suitable for high school students in Grades 9-12.

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