The equation \(\left(x-3\right)^2 = -56\) involves a square on the left-hand side that is set equal to a negative number. Let's solve this step-by-step.

### Step 1: Recognize the problem with the square root
The square of any real number is non-negative. Therefore, an equation like \(\left(x-3\right)^2 = -56\) has no solution among real numbers because no real number squared equals a negative number. This indicates that the solutions involve complex numbers.

### Step 2: Solve the equation for complex numbers
To find the solution in the complex number system, take the square root of both sides:

\[
\sqrt{\left(x-3\right)^2} = \sqrt{-56}
\]

This simplifies to:

\[
x - 3 = \pm \sqrt{-56}
\]

### Step 3: Simplify the square root of a negative number
The square root of \(-56\) can be written as:

\[
\sqrt{-56} = \sqrt{56} \cdot i
\]

So the equation becomes:

\[
x - 3 = \pm \sqrt{56} \cdot i
\]

### Step 4: Solve for \(x\)
Add 3 to both sides to solve for \(x\):

\[
x = 3 \pm \sqrt{56} \cdot i
\]

### Final Answer:
The solutions to the equation are:

\[
x = 3 + \sqrt{56} \cdot i \quad \text{or} \quad x = 3 - \sqrt{56} \cdot i
\]

### Simplified Radical Form
The square root of 56 can be simplified further:

\[
\sqrt{56} = \sqrt{4 \times 14} = 2\sqrt{14}
\]

Thus, the final solutions are:

\[
x = 3 + 2\sqrt{14} \cdot i \quad \text{or} \quad x = 3 - 2\sqrt{14} \cdot i
\]

Would you like more details on any part of the solution, or do you have any questions?

### Follow-Up Questions:
1. What are complex numbers and how are they used in equations like this?
2. How can the square root of a negative number be interpreted geometrically?
3. What does the \(i\) in the solution represent?
4. How do you simplify square roots like \(\sqrt{56}\)?
5. Could there be a graphical interpretation of this solution on the complex plane?

### Tip:
When dealing with quadratic equations, always check the discriminant (\(b^2 - 4ac\)) to determine the nature of the roots (real vs. complex).

Explore how to solve the equation (x-3)^2 = -56 using complex numbers. This problem delves into complex number operations and the interpretation of square roots of negative numbers in equations. Suitable for advanced high school students.

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