2. \( x^{2}+15 x=-50 \)

Let's take a quick stroll through history! Quadratic equations, like the one you've provided, date back to ancient Babylonians around 2000 BC, who used geometrical methods to solve them. These early mathematicians didn’t have the symbolic notation we do today, yet they mastered solving problems involving areas and squares! Fast forward to the 17th century, where mathematicians like René Descartes and François Viète began to formalize our understanding of these equations, introducing techniques that remain crucial even in modern algebra. Now, onto practical applications! Quadratic equations pop up everywhere in the real world, from physics to business. For instance, when you throw a ball, its path can be described using a quadratic equation. Understanding how changes in velocity and angle impact the ball's trajectory can help athletes improve their performance. Similarly, businesses can use quadratics to determine profit maximization by analyzing cost versus revenue scenarios. So, next time you see a parabolic arc, remember—there's a quadratic lurking behind it!