A particle with total energy \(3.5 \mathrm{J}\) is trapped in a potential well described by \(U=7.0-8.0 x+1.7 x^{2},\) where \(U\) is in joules and \(x\) in meters. Find its turning points.

The potential energy equation in physics underpins the behavior of a particle within a force field, such as a gravitational or an electromagnetic field. It mathematically encapsulates how the position of a particle relates to the energy stored due to its position or configuration. One standard form of potential energy that arises in classical mechanics is the quadratic potential, often representing the energy in a harmonic oscillator or any system that can be approximated as such.

For instance, in our exercise, the potential well is described by the equation \(U=7.0-8.0x+1.7x^{2}\). To find the turning points of a particle, meaning where it changes direction in the well, we set the total mechanical energy \(E_{total}\) equal to the potential energy \(U\). This results in the particle's kinetic energy being zero at these points, as it momentarily comes to rest before reversing direction. By doing so, we create an equation that captures the relationship between the position of the particle and the potential energy at those critical points.

In more complex scenarios, potential energy can have different forms, such as gravitational potential energy expressed by \(U = mgh\), where \(m\) is mass, \(g\) is gravitational acceleration, and \(h\) is height. Understanding the potential energy equation is crucial for predicting the motion of particles in physical systems.

The quadratic formula is a critical tool in physics, as it offers a way to solve quadratic equations, which frequently appear in various physical contexts. A quadratic equation is characterized by the standard form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. The quadratic formula is the solution to this form of equation and is given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

In our original exercise, the equation obtained from equating the potential energy to the total energy of the particle results in such a quadratic equation. By substituting the corresponding values into the quadratic formula, we can solve for the values of \(x\), which are the positions corresponding to the turning points of the particle in the potential well. These solutions are crucial they indicate where the particle will stop and reverse its course.

Understanding the quadratic formula and how to apply it in physics is not only essential for finding the roots of quadratic equations but also for comprehending the physical implications of these roots—for example, interpreting them as positions, times, or energies depending on the context of the problem.

Solving for roots in physics is a fundamental aspect of understanding the motion and equilibrium points within physical systems. Roots of an equation represent the values for which the equation holds true. In mechanics, these can correspond to positions of stability or transition, such as the turning points in the potential well as illustrated by our textbook problem.

The process of finding these roots often involves rearranging the original equation to a recognizable form and then applying mathematical techniques to derive the solution. When dealing with quadratic equations, as detailed in our exercise, we typically resort to the quadratic formula for precise answers. It's important to note that not all roots may have physical meaning. For instance, in certain contexts, negative roots might be dismissed if only positive quantities make sense. Moreover, complex roots (involving imaginary numbers) usually indicate oscillatory motion, common in wave phenomena and alternating currents in electrical circuits.

In learning to solve for roots within physics problems, students enhance their problem-solving abilities and deepen their understanding of the natural world. It's skills such as these that enable physicists and engineers to design systems and predict behaviors that align with the underlying laws of the universe.