Which of the following best describes resolution?

The best description of resolution in this context is the concept of a firm decision to do or not to do something. Resolution often refers to a commitment or determination to follow through on a specific action or change, especially in contexts like New Year's resolutions, where individuals make pledges to improve themselves or alter habits. This emphasizes a sense of purpose and determination, showcasing that resolution involves not just the intention but also the commitment to actualize that intention into action.

The other options do have their own meanings but do not encapsulate the essence of resolution as effectively. For instance, while solving a mathematical equation involves a process, it lacks the personal commitment aspect tied to resolution. A hopeful outlook towards the future suggests optimism but doesn't relate directly to the idea of making a decisive choice. Similarly, a method used to achieve a personal goal is more about strategies or techniques, rather than the firm decision-making characteristic that defines resolution.