Laptop will not boot, get 7 beep error code. Need more information.

A few months ago my Inspiron 1546 started hanging in the Dell boot screen. This progressed to it freezing in that screen and eventually just turning itself off after 5 or so minutes. It would always eventually boot though, just took a few tries, until a couple weeks ago it stopped booting altogether. If I power it on, the fan goes, and the optical drive is reading, but I get no picture, and it just turns itself off. If I boot with FN+Power I get a 7 beep error code. I tried troubleshooting by: exhausting flea power, removing peripheral devices including checking RAM sticks one at a time, booting with no battery and just the power cord, resetting the CMOS battery, and connecting to an external monitor (I have seen people suggest this...).

I let the laptop sit for a couple weeks as I was busy, and yesterday got it back out again to try to figure out what's wrong. This time I booted it like normal and got the same black screen and then power off. However, this time when I FN+Power booted, it loaded diagnostics, and when they were finished everything checked out, and it asked for the HD, which I installed and then booted the computer like normal. I used the computer for about 8 hours, and ran some diagnostics and during that time everything was 100% fine. The battery was dead when it powered on and it recharged fully while I used the laptop. I restarted it and it again booted like normal. After I shut it down and came back the next morning, I got the black screen and 7 beeps again from FN+Power.

I am at a loss for what is wrong with the laptop. I am out of warranty, but I am looking for advice on something to try next. I know the 7 beep error code signifies bad CPU, but it seems odd that it boots fine and the CPU tested excellent. Maybe it had something to do with the dead battery or AC power on the board? Any input would be greatly appreciated.