Some stains, the only way to get them out, is to cut them out. Burn marks, bleach, or stubborn stains are not pleasant to look at. Cutting up your carpet isn't as scary as it sounds.
To patch your carpet we'll cut a square around the ruined section of carpet, cut the same sized square out of an unused section of carpet, then patch the new piece of carpet in by attaching it to the carpet around it. This allows the patch to move with the rest of the carpet. When we're all done, we run a roller over it, causing the carpet fibers to interlock, and make your patch look like just a normal part of the carpet.