3) \( a^{2}+14 a-51=0 \)

To solve the quadratic equation \( a^{2} + 14a - 51 = 0 \), you can use the quadratic formula, which is \( a = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \). Here, \( a = 1 \), \( b = 14 \), and \( c = -51 \). Plugging these values in gives \( a = \frac{-14 \pm \sqrt{14^2 - 4 \times 1 \times (-51)}}{2 \times 1} = \frac{-14 \pm \sqrt{196 + 204}}{2} = \frac{-14 \pm \sqrt{400}}{2} = \frac{-14 \pm 20}{2} \). This simplifies to two possible solutions: \( a_1 = 3 \) and \( a_2 = -17 \). Understanding quadratic equations dates back to ancient civilizations like the Babylonians, who explored these mathematical relationships. They didn't have our modern symbols, but they managed to solve quadratic-like problems using geometric methods. It’s fascinating to think about the evolution of math, isn’t it? Quadratic equations pop up everywhere! From determining the trajectory of a basketball to calculating profit maximization in business, knowing how to solve them can be a game-changer in real life. Just imagine using this knowledge to help smooth over that awkward moment in class when the teacher asks for volunteers to solve math problems!