Remove the old carpet pad.
Replace section of carpet pad.
A line or 2 of glue on the surface of the carpet pad will often be sufficient.
Using a razor knife cut around the section of padding you are removing so you can pull it away from the rest of the padding.
If carpet tape is not doing the job you can also use carpet glue to hold the patch in position.
Press it over the damaged area and turn until you reach the padding then pull out the circular piece and patch as you would a large section.
Keep in mind that if you plan on replacing the carpet the patch will adhere to the padding making it harder to salvage the pad.
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Older carpet in high traffic areas might be more faded.