If I need to sleep nine hours, and wake up at 6:45, what time should I be going to bed?
You should go to bed at 9:25. This will give you 20 minutes to rest and relax to wind down prior to falling asleep at 9:45.
You need to sleep at 9:45 pm the previous night to wake up at 6:45 having enjoyed 9 hours of sleep. From 9:45 PM to 9:45 AM is 12 hours, then minus the three hours to get back to 6:45 AM.
9:30 p.m. if you're a Snoozer like I am. I hit that snooze button at least twice :)