2. Solve for x using the quadratic formula 1.2.1. \( x^{2}+8 x+15=0 \)

To solve the quadratic equation \( x^{2}+8 x+15=0 \) using the quadratic formula, we first identify the coefficients: \( a = 1 \), \( b = 8 \), and \( c = 15 \). Plugging these values into the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) gives us \( x = \frac{-8 \pm \sqrt{8^2 - 4(1)(15)}}{2(1)} \). Simplifying this leads to \( x = \frac{-8 \pm \sqrt{64 - 60}}{2} = \frac{-8 \pm \sqrt{4}}{2} = \frac{-8 \pm 2}{2} \). So the solutions are \( x = -3 \) and \( x = -5 \)! Encouragingly, this formula came about in the 9th century, introduced by the mathematician Al-Khwarizmi. His work laid the foundation for algebra, making it not just a tool for solving equations, but also an exciting intellectual adventure! Isn't it fascinating how ancient insights still help us tackle math problems today?