2. \( x^{2}-3 x-10=0 \)

To solve the quadratic equation \( x^{2}-3 x-10=0 \), we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a} \), where \( a=1, b=-3, \) and \( c=-10 \). Plugging in these values, we find \( b^{2}-4ac = (-3)^{2} - 4(1)(-10) = 9 + 40 = 49 \). Thus, \( x = \frac{3 \pm 7}{2} \), leading to the solutions: \( x = 5 \) and \( x = -2 \). Quadratic equations have been around for thousands of years and were studied by ancient Babylonian mathematicians who had methods for solving them—even if they didn’t express it in the same way we do today! This blend of geometry and algebra set the stage for much of modern mathematics, making it truly fascinating how these principles have stood the test of time. In the world of engineering, physics, and economics, quadratic equations pop up all the time! Whether it’s determining the optimal trajectory of a projectile or maximizing profit in a business model, understanding how to solve these equations can lead to results that positively impact real-world decision-making. So next time you spot a quadratic, think of it as a key to unlocking practical solutions!