The equation $(x - 5)^2 = -9$ has no real solutions because the left side, $(x - 5)^2$, represents a squared term, which is always non-negative for real numbers. Squaring any real number results in zero or a positive number, so it cannot equal a negative number like $-9$.

To confirm, let's analyze this:

1. Expanding or taking the square root of both sides doesn't change the fact that $(x - 5)^2 \geq 0$.
2. Since there’s no real value for $x$ such that $(x - 5)^2 = -9$, we conclude that there are no real solutions.

However, if we consider complex numbers, we can proceed by taking the square root of both sides:

$x - 5 = \pm \sqrt{-9}$

Since $\sqrt{-9} = 3i$ (where $i$ is the imaginary unit), we have:

$x - 5 = \pm 3i$

Solving for $x$, we find:

$x = 5 \pm 3i$

Solution

The solutions in the complex number system are: $x = 5 + 3i \quad \text{and} \quad x = 5 - 3i$

Would you like further details, or have any questions?

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Here are five related questions to expand on this concept:

1. How do you solve equations involving imaginary numbers?
2. What is the process for expanding and simplifying $(x - 5)^2$?
3. Why is it impossible for real numbers to have a square root of a negative value?
4. How can complex solutions be graphed in the complex plane?
5. What are the real-world applications of complex numbers?

Tip: When you encounter a negative square, consider if complex numbers may be needed for solutions.