Re: I hate
I came across a thread about this in a search a while back when looking for something else. Maybe do a search for "burned carpet" or similar. There were 2 links posted for DIY carpet repair. One mentioned exactly as you did by using a piece from the back of a closet for larger spots. It also talked about slightly trimming the top black burn marks with a scissors. If done well you may be able to "erase" those remaining marks.
If all else fails they say the truth will set you free, but I'm not sure who "they" are.