motherboard

You need to have Electronic Engineering Skills and proper equipment like an Oscilloscope , Signal injector, Voltage Regulator and a service manual on the equipment itself to do logical analysis and reverse engineering of it  to locate the trouble.

Wes

There are things that you can do to troubleshoot it to see if it is possible it is the motherboard.  Laptops (just like desktops) have parts that you can change out....and replace, let alone just completely remove to test.

IE:  Try replacing the ram with known good ones.  Pull the Hard drive and Optical drive or cd rom drive, floppy drive etc...(dont need any drives to make the laptop turn on).  Remove the battery (dont need it to turn it on if your power cord is good and functional.)
Then try turning the laptop and if it boots, most likely it is something else and not the motherboard that is bad.  Also you can sometimes replace the CPU as well...so again if you have a known good one that will work for that SPECIFIC Model of laptop, then you can try replacing that to see as well.  
Otherwise is it possible to repair the motherboard....YES....but is it easy?  without knowledge of what is causing the board to go bad and without diagnostics or testing equipment as wmiller has stated....or even Laptop motherboard diagnostic tools (there are some out there....)not easy for one that has know skill set in doing so.

So again you can try some of the things I suggested to Determine if the board is bad or some other piece....or get diagnostic tools to test.