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The equation provided is:

\[
\sqrt{x - 27} = -\sqrt{x + 9}
\]

### Step-by-Step Solution:

1. **Isolate the square roots**:
   \[
   \sqrt{x - 27} = -\sqrt{x + 9}
   \]
   This equation states that the square root of a number is equal to the negative of another square root. However, this is problematic because the square root function always results in a non-negative value, and there is no real number for which a square root is negative.

2. **Conclusion**: 
   This equation has **no solution** because the left-hand side is always non-negative (as it's a square root), while the right-hand side is always non-positive due to the negative sign. There's no value of \( x \) that can satisfy this condition.

Thus, the correct answer is **No solution**.

---

Do you have any questions, or would you like further details? 

Here are 5 related questions to think about:
1. What are the properties of square roots that influence their solutions?
2. How do we handle equations that involve both positive and negative square roots?
3. Are there any situations where an equation like this could have a complex solution?
4. How would this equation change if the negative sign were removed from the right-hand side?
5. What other types of functions share similar restrictions as the square root function?

**Tip**: Always check the basic properties of functions like square roots (which are non-negative) when solving equations, as this can help you avoid extraneous solutions or identify contradictions early.

Solve for x, where x is a real number: √(x - 27) = -√(x + 9)

Learn how to solve the radical equation √(x - 27) = -√(x + 9). This algebraic problem highlights important concepts such as the properties of square roots and why the equation has no real solution. Suitable for high school students in Grades 10-12.

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