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I gave an answer that wasn't well received (my answer has a negative score).

I plan to research my answer to see if the objections are valid. (I want to check if the geolegs are as precise as I think they are)
If I don't agree with the critics, I will respond. If they are valid, would it be appropriate to delete my answer? Do I somehow indicate that I am changing my mind and that's why I am deleting it? If I delete it, will the negative points be deleted as well?

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It can be appropriate to self-delete content. One criterion one can use is if one believes that the post adds something useful to the site. If you feel it does not, it can make sense to delete it. However, if you feel there is still a relevant point there, it can make sense to keep it even at negative score.

As deletions are soft-deletions only, and the content stays to some extent visible for some users, it can make sense to explain why the post was deleted, to avoid confusion. (But it is not a necessity either.) Posting an explanatory comment on the post before deleting it is a way to do this. In this situation, a comment is often preferable to an edit as the edit will bump the thread and the intent when deleting is typically not to draw extra attention to the deleted content.

When content is deleted usually (there are some exceptions for old content) the votes on it are deleted, too, and the global point-count is adjusted accordingly. Yet note that at the time of writing you have one up-vote and two down-votes on this post, for a net-effect of $1 \times 10 + 2 \times (-2)= 6$ so that deleting the post would rather lose some points.

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