Let's start by solving the given equation:

\[
\sqrt{-2x + 3} = -8 + \sqrt{5x + 76}
\]

### Step 1: Analyze the equation
First, notice that the square root of any real number is always non-negative. Therefore, the expression \(\sqrt{-2x + 3}\) must be non-negative, but the right-hand side involves \(-8\), which makes the expression on the right negative. This suggests that the equation doesn't have a real solution because a square root can't equal a negative number.

### Step 2: Check the domains of the square roots
For \(\sqrt{-2x + 3}\) to be valid, the expression inside the square root must be non-negative:
\[
-2x + 3 \geq 0
\]
\[
x \leq \frac{3}{2}
\]

For \(\sqrt{5x + 76}\) to be valid:
\[
5x + 76 \geq 0
\]
\[
x \geq -\frac{76}{5}
\]

### Step 3: Conclusion
Since the equation \(\sqrt{-2x + 3} = -8 + \sqrt{5x + 76}\) involves setting a square root expression equal to a negative number, there is no real solution.

Would you like more details on this, or have any other questions?

Here are 5 related questions you might find interesting:
1. What is the process for solving equations involving multiple square roots?
2. How can you verify the domain of square root functions?
3. What are extraneous solutions, and why are they important in solving equations with square roots?
4. Can complex solutions exist for equations involving square roots?
5. How do inequalities involving square roots work?

**Tip:** Always check the domain of square root expressions to avoid unnecessary steps in solving!

Solve the equation: \( \sqrt{-2x + 3} = -8 + \sqrt{5x + 76} \)

Explore the solution to the equation \( \sqrt{-2x + 3} = -8 + \sqrt{5x + 76} \), where we analyze domain restrictions and the properties of square root functions to conclude that no real solutions exist. Suitable for students in Grades 9-12.

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