\( \frac { - 8 \pm \sqrt { - 128 } } { 2 } \)

Let's break that down a bit! The expression \( \frac { - 8 \pm \sqrt { - 128 } } { 2 } \) features a square root of a negative number, which points us straight to the land of imaginary numbers! The \(-128\) can be rewritten as \(128i^2\) (because \(\sqrt{-1} = i\)), leading us to \(\sqrt{128} \cdot i\), where \(\sqrt{128} = 8\sqrt{2}\). So, we can simplify our expression to: \[ \frac{-8 \pm 8\sqrt{2}i}{2} = -4 \pm 4\sqrt{2}i \] And voilà! You’ve got two complex solutions. 🌈 Imaginary numbers aren't just fun math concepts; they have real-world uses! For example, they come into play in electrical engineering, particularly in analyzing alternating current (AC) circuits, where the properties of voltages and currents can be expressed using complex numbers. This allows engineers to handle phase differences and impedance in a much more straightforward way than using real numbers alone. Isn't that neat? ⚡