\( 10 z ^ { 2 } + 17 \equiv + 7 \)

Let's simplify the equation step by step! First, we can rearrange it to make it clearer: \( 10z^2 + 17 - 7 \equiv 0 \). This simplifies to \( 10z^2 + 10 \equiv 0 \), or further \( 10(z^2 + 1) \equiv 0 \). This means \( z^2 + 1 \equiv 0 \), leading to \( z^2 \equiv -1 \). In real numbers, this doesn't have a solution, but in the complex plane, we find that \( z \equiv i \) or \( z \equiv -i \)! Now, let's have some fun: Imagine you're a detective of numbers! When you first encounter \( z^2 + 1 \equiv 0 \), it looks a bit suspicious, like a clue that leads you into the mysterious world of imaginary numbers. You’re on the trail of that elusive \( i \), the imaginary unit, who dances under the arch of -1. And while real numbers might play it safe, the complex numbers throw a wild party with all sorts of twists and turns!