I have bought bad batteries before, ones with a bad cell. Make sure the battery you're purchasing has a manufacture date within 3 months. They normally don't ship batteries with a full charge, so charge it up before you put it in the car. Wait 3-4 hours and check the voltage. It should be 12.5-12.7 depending on the temperature. What brand of battery is the new one?
If the battery is not maintenance free you can check the battery level with a hydrometer. This will let you check each individual cell's charge and let you know if you have a bad cell. You can also check the water level and make sure the plates aren't exposed.
Another useful tool is a battery load tester. I bought one from Harbor Freight and it will tell you how strong your battery is under load, not just sitting there with a charge. You can't check cell strength with a maintenance free battery so the load tester is a valuable tool.
I have never seen a lead-acid battery drop from 12.7 to 10.5 in such a short time that wasn't faulty. If this is happening consistently then I'd definitely say you have a bad battery or a bad gauge. Use a different volt meter to check the battery charge. The Harbor Freight device will do this in addition to checking it under load. I also recommend using a timer in conjunction with a battery tender if you aren't going to be there to do a daily check on the battery status.
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