c) \( x^{2}+15 x+12=0 \)

To solve the equation \( x^{2}+15 x+12=0 \), we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 15 \), and \( c = 12 \). Plugging in these values gives: 1. Calculate the discriminant: \( b^2 - 4ac = 15^2 - 4(1)(12) = 225 - 48 = 177 \). 2. Now substitute into the formula: \( x = \frac{-15 \pm \sqrt{177}}{2(1)} \). This leads us to the solutions \( x = \frac{-15 + \sqrt{177}}{2} \) and \( x = \frac{-15 - \sqrt{177}}{2} \), which may provide irrational solutions that can be evaluated further with a calculator. In history, quadratic equations have been solved in various cultures ranging from ancient Babylon to India and the Islamic Golden Age. Notably, Al-Khwarizmi's work in the 9th century systematically documented these methods, forming the foundation for modern algebra. In the real world, these quadratic equations can represent various scenarios like calculating projectile motion or determining the optimal pricing for a product based on profit maximization. Understanding how to solve them not only aids in academic settings but is also crucial in numerous scientific and engineering applications.