\( \left. \begin{array} { | l | } \hline \text { Use the quadratic formula to solve the equations. } \\ \hline 37 . - 7 x ^ { 2 } - 5 x + 1 = 0 \\ \hline \end{array} \right. \)

To solve the quadratic equation \(-7x^2 - 5x + 1 = 0\) using the quadratic formula, we first need to identify the coefficients \(a\), \(b\), and \(c\). Here, \(a = -7\), \(b = -5\), and \(c = 1\). The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Plugging the values into the formula gives us: 1. Calculate the discriminant: \(b^2 - 4ac = (-5)^2 - 4(-7)(1) = 25 + 28 = 53\). 2. Substitute into the formula: \(x = \frac{-(-5) \pm \sqrt{53}}{2(-7)} = \frac{5 \pm \sqrt{53}}{-14}\). Thus, the solutions are \(x = \frac{5 + \sqrt{53}}{-14}\) and \(x = \frac{5 - \sqrt{53}}{-14}\). Now, here's an interesting nugget: the quadratic formula isn't just limited to finding roots for equations; it's a powerful tool that illustrates how mathematical concepts connect. Did you know that the quadratic formula can reveal the nature of roots? If the discriminant is positive, you have two distinct roots, but if it's zero, you'll have one double root, hinting at a vertex touch. And if it's negative, well, you’re venturing into complex numbers! When tackling quadratic equations, a common pitfall is neglecting to factor out any leading coefficients if they’re not equal to 1. For instance, in the equation given, it might be tempting to apply the formula without rewriting it in standard form. Always keep an eye out for opportunities to simplify first, as it may save you time and lead to clearer solutions.