Use your utility knife to cut around the burnt section. Cut through the carpet backing but not the carpet padding.
Use the burnt piece as a template to cut your new replacement carpet. Remember to match any pattern and the direction of the carpet's nap.
Cut out a similarly shaped piece of carpet tape that has a diameter 2 inches larger than your carpet hole. Peel the backing of the tape. Slip the tape between the carpet backing, sticky side up, so that it extends beyond the hole.
Trim your replacement piece so it fits as perfectly as you can make it. Place the new piece on the carpet tape. Press firmly down on the new piece, making sure all edges align.
Trim the edges of the new piece so it matches the length of the original. Place your heavy books on top of the new piece for several hours so it can fully blend into the original. Use the carpet tractor to fully blend the new carpet with the old.