If the patch didn't work, it could be a poor patch job. Bad prep work, not enough adhesive, or not rolled out with enough force. A good patch job would include a rope seal shoved into the puncture, then trimmed flush with the inside of the tire and finally the patch applied with it being centered over the puncture.
Many tire stores don't check a patch after they install one and remount the tire on the wheel. The tire and wheel should have been inflated to about 40psi and put into a water tank to check for any other leaks.
You can do some leak checking by taking the wheel off the car and squirting a mix of dish soap and water from a squirt bottle around where the puncture was to see if any bubbles form. Do the entire surface of the tire to see if there is another puncture.
Also spray around the rim to check for any leaks. Sometimes you can have material on the rim and the tire bead that can not allow the tire to seat properly. Check the valve stem base and the valve itself. Anywhere you see bubbles forming, that's your leak.
Tires can be expensive so it really depends on the tread remaining on the tire. I would not replace just one tire if there was 50% or so tread remaining on the other tire. It's always a good idea to replace tires in pairs. Tires can run anywhere from $200 to $500 for a pair depending on brand, model and size. The 275/40 and 285/40 sizes are usually within a few dollars of one another.
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