If carpet tape is not doing the job you can also use carpet glue to hold the patch in position.
Replace section of carpet padding.
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Older carpet in high traffic areas might be more faded.
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.
Remove the old carpet pad.
Continue around the other edges of the patch in the same manner except in the case where an edge meets a wall as mentioned above.
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.
From the removal and disposal of your old carpet to laying your new carpet and padding the local installation team will make sure the job is done right and to your satisfaction.
If you are only replacing a 6x9 foot area with no seams i would likely just replace the carpet and pad if i could find a usable pattern on roll at lowes or home depot or a remnant at a dealer.
In either event be prepared to vacuum up a lot of dirt and possible degraded carpet pad.
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.
This includes helping you select quality padding to install under your new carpet to help it last longer feel softer reduce noise and provide insulation.
A line or 2 of glue on the surface of the carpet pad will often be sufficient.