(x-8)^2=-8

2 min read Jun 17, 2024

Solving the Equation (x - 8)² = -8

This equation presents a unique challenge because we're dealing with a squared term resulting in a negative value. Let's break down the steps to solve it:

Understanding the Problem

Squaring a number always results in a non-negative value. This means that no real number, when squared, can equal -8.
Therefore, there are no real solutions to this equation.

Exploring Complex Numbers

To find solutions, we need to introduce the concept of complex numbers. These numbers extend the real number system by including the imaginary unit i, where i² = -1.

Solving with Complex Numbers

Take the square root of both sides: √((x-8)²) = ±√(-8)
Simplify: x - 8 = ±√(-8) = ±√(8)√(-1) = ±2√2 * i
Isolate x: x = 8 ± 2√2 * i

The Solutions

The solutions to the equation (x - 8)² = -8 are:

x = 8 + 2√2 * i
x = 8 - 2√2 * i

These are complex numbers, where the real part is 8 and the imaginary part is ±2√2.

Conclusion

While the equation (x - 8)² = -8 has no real solutions, it does have two complex solutions. This highlights the importance of understanding complex numbers and their applications in solving equations that involve negative squares.