On Mon, 26 May 2014 03:28:54 +0000 (UTC), Calia
wrote:

I have a Windows 7 HP G72 laptop which suddenly stopped working.

Jumping to a premature conclusion, I would guess(tm), that you have
the same problem as all these people:
https://www.google.com/#q=hp+g72+laptop+will+not+turn+on
Usually, it's bad BGA chip soldering.

Do you have an idea how to do basic troubleshooting?

Unplug the charger. Remove the battery. Go away for about 10
minutes. Put the battery back in and do not plug in the charger. If
the battery is totally dead (i.e. shorted from sitting at zero charge
for too long), it might not turn on. If you suspect that you have a
shorted battery, and can't find a suitable replacement, remove the
battery, insert the charger, and see if it will power on with just the
charger.

What happens when I try to reboot is that nothing shows
up on the 17-inch screen whatsoever.

That might be an HP G72-227WM. Is that correct? (Complete and
accurate model numbers are always helpful).

The only indicators are the fan and a half dozen LEDs.

The fan is most notable.
When I press the power switch, the fan goes on for what
I timed at 40 seconds. Then the fan goes off.

Ummm... It should go at full blast for about 2 seconds, and then spin
slowly. 40 seconds implies that some timer on the MB is not working
normally.

The white LED to the immediate left of the topside power
switch glows white when the computer is turned on.

OK, the power switch is working. Is there anything on the screen? If
not, take a flashlight and shine it around the upper left of the
screen, where the BIOS boot messages appear, or in the center of the
screen, where the Windoze logo normally appears. If you see
something, then the backlighting is blown. The G72-227WM has an LED
backlight, so there's no inverter to replace. The problem will be to
determine if the lack of backlighting is caused by a problem on the
motherboard, LCD panel, cable, or connector.

The orange and white LED next to the power input on the
right side of the pc also glows either orange or white.

White means it has power. Orange means it's charging. No light means
that it's charged. Flashing means your battery is so low as to not be
able to power the laptop.

The white capslock key LED blinks white synchronously with
the NumLock key. The F12/WiFi function key glows orange.

HP Blinkenlighten codes:
http://h10010.www1.hp.com/ewfrf/wc/document?lc=en&dlc=en&cc=us&docname=c01732674&prod uct=1132551#N213
This should give you some things to check.

The tiny LED on the DVD tray button does not light, nor
will the DVD tray open unless I insert a paperclip.

I found this "maintenance guide" but it doesn't seem to
list these conditions:
http://h10032.www1.hp.com/ctg/Manual/c02623152.pdf

That's meant for users, not techs.

Those of you with experience troubleshooting a dead
laptop, can you provide advice?

Yes, I can. Try asking in the HP support forums:
http://h30434.www3.hp.com/t5/Notebook-Hardware/bd-p/Hardware
Hint: You'll need to provide a better description of the flashing
lights and boot errors.

Mainly I just want to know if I can pinpoint the failure
and then I can see whether it's worth trying to fix.

In my never humble opinion, the last few years of HP and Compaq laptop
are not worth fixing. I hesitate to fix them because they don't stay
fixed. I often see them back in a few months, usually with a
different problem. Details and horror stories on request.

--
Jeff Liebermann
150 Felker St #D http://www.LearnByDestroying.com
Santa Cruz CA 95060 http://802.11junk.com
Skype: JeffLiebermann AE6KS 831-336-2558