a) \( x^{2}-5 x-14=0 \)

To solve the quadratic equation \( x^{2}-5 x-14=0 \), you can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -5 \), and \( c = -14 \). Plugging in these values gives you \( x = \frac{5 \pm \sqrt{(-5)^2 - 4(1)(-14)}}{2(1)} \). This simplifies to \( x = \frac{5 \pm \sqrt{25 + 56}}{2} = \frac{5 \pm \sqrt{81}}{2} = \frac{5 \pm 9}{2} \). So, the solutions are \( x = 7 \) and \( x = -2 \). Quadratic equations like this date back to ancient civilizations such as Babylon, where early mathematicians had their own methods for solving these problems without the formal notation we use today. It's fascinating how this foundational concept in algebra has evolved over centuries and still forms the basis for many mathematical applications today! In real life, quadratic equations often pop up in situations like projectile motion, where the path of an object in the air can be modeled by a quadratic function. Whether you’re launching a basketball or calculating the trajectory of a rocket, understanding how to solve these equations can help predict outcomes and optimize performance!