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They may care less with time, but still remember. This doesn't mean fixing your reputation is impossible. Bad reputations are always potentially temporary. It all has to do with time. After some time, people stop caring so much about what you did in the past. Many people encounter their own failures and the failures of others over time. These stand out more, and they forget about you. Remember that you remember your own failures better than other people. Your bad reputation may be worse in your head than it is in the eyes of others. Research shows that, contrary to popular perception, when compared to other means of social evaluation, reputation matters measurably less than other factors. Go off the grid and take a break from social media and online. Be frank when addressing the issues surrounding your bad reputation. If you are calm, respectful, and genuinely concerned, they are likely to help you. By not verbally addressing the problem, you let people's imaginations run wild, and you give rumors free-range. If someone is actively spreading news of your bad reputation, talk to them first. Ask friends their opinion on the matter. Try not to be defensive. Maybe your bad reputation is a superpower in disguise. Take some time to think about how your bad reputation may be a positive thing and reframe your perception of the situation. You might not be promiscuous, you could be in charge of your sexuality. If people think you play a sport too aggressively, maybe you're your team's heavy artillery. Sometimes you just need to control the perceptions of others. They may not be seeing the situation in the right light. If you can, help others to see the situation in an entirely different light. Provide a calculated positive perspective. This means finding ways to reframe events and actions that give you a bad reputation. For example, sometimes people shame promiscuous behavior, while others present promiscuity as "owning your sexuality" or being "sex positive." Think of ways you can help people see your actions in a new light. Be aware that you may not be able to sway everyone's opinions. You may just need to say the reputation is not true. If you have been lied about, it may take some time to undo the damage. Be assertive (not aggressive), and make sure you spread the word as soon as possible. Ask other people to help set the story straight. The longer this bad reputation exists, the harder it may become to manage it. Clarify any misunderstandings. Be vocal when challenging lies.
Accept that people might not forget. Talk to people about it. Turn it into a strength. Change how others think. Denounce accusations as lies.