d) \( x^{2}-5 x-6 \)

To solve the quadratic equation \( x^{2} - 5x - 6 = 0 \), we can factor it. We need two numbers that multiply to -6 (the constant term) and add to -5 (the coefficient of the x term). These numbers are -6 and 1. Thus, we can write the equation as \( (x - 6)(x + 1) = 0 \). Setting each factor to zero gives us the solutions: \( x = 6 \) and \( x = -1 \). Quadratic equations like this one have been essential throughout history, especially during the time of the ancient Greeks who first explored geometric solutions to such equations. They laid the groundwork for what we now refer to as algebra! In real-world applications, finding the roots of a quadratic equation can help in various fields like physics, engineering, and even finance, where you might need to determine the time of flight of a projectile or optimize profit scenarios. Understanding how to solve such equations opens doors to diverse real-life problem-solving scenarios!