Psychic powers diagnosis: motherboard failed.
More logical approach: Minimize the number of variables. Inspect fans during powerup, check for spinning or pulses. Remove and set aside battery, hard disk. If possible, test all removable parts in another laptop, ideally one of same/similar model.
Depending on the client involved, just being able to triangulate and price the failed parts is all most folks are willing to do. Putting the old laptop's HDD in a USB enclosure and helping them shop for a replacement laptop usually gets you full fee, gratitude, and often rights of salvage to whatever's still usable.
Behemoth - media server
Galactica - test lab server
Falling Outside the Normal Moral Constraints - MBP 15
Donnager - M4800 DTR
Rocinante - E5530 testing machine
Saratoga, Ticonderoga, BunkerHill - the Hydra desktop PC-thing.