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

I have a Windows 7 HP G72 laptop which suddenly stopped working.
Do you have an idea how to do basic troubleshooting?

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

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.

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

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

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

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

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

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