## Solving the Equation: (k-4)^2 = -3

This equation presents a unique challenge because it involves a squared term equaling a negative number. Let's break down how to solve it and understand the implications.

### Understanding the Basics

**Squaring a number always results in a non-negative value.**This means that any real number multiplied by itself will always be zero or positive.**Therefore, a perfect square cannot equal a negative number.**

### Solving the Equation

Given the equation **(k-4)^2 = -3**, we know that there are **no real solutions** for the variable 'k'. This is because the left side of the equation can never be negative, while the right side is fixed at -3.

### Exploring Complex Numbers

To find solutions to this equation, we need to delve into the realm of **complex numbers**. Complex numbers are an extension of real numbers, including the imaginary unit 'i', where **i² = -1**.

Let's rewrite our equation:

(k-4)² = -3

Taking the square root of both sides:

k - 4 = ±√(-3)

Simplifying:

k - 4 = ±√3 * √(-1)

k - 4 = ±√3 * i

Solving for 'k':

k = 4 ± √3 * i

### The Solution

Therefore, the solutions to the equation (k-4)² = -3 are **complex numbers**, specifically:

**k = 4 + √3 * i****k = 4 - √3 * i**

### Conclusion

While the equation (k-4)² = -3 does not have real number solutions, it does have two **complex solutions**. This demonstrates that the realm of complex numbers allows us to solve equations that would be impossible to solve within the set of real numbers.