Motherboards are way too complex for a human to actually fix....it would be possible, but it would take forever.
Most motherboards these days are at least 8 layers of silicon with gold/metal traces inside of them. Even if 1 trace is damaged it could possibly kill a motherboard. A capacitor might be easier to fix...but you would need to find the correct type, and would require fine soldering tools.
So plasmastorm is correct. You can only replace them...
gonna call BS. if you take the time to learn how to solder/take apart your laptop to get to the mobo, learn how to use a multimeter correctly and then are able to either see (sometimes its more obvious than you may think) or locate issues via multimeter. you can attempt to replace/bridge fuses. caps and other things if needed. especially if you are at the point to where your laptop isnt ever going to work until you either pay to replace it or are able to repair the motherboard, its not like you have anything to lose. just b to careful not to do more damage while you make the rounds. i have (not without a good amount of luck in a few cases im sure) been able to replace fuses that could only be removed after shaving/filing my soldering iron to a razor point and tweezers but can say and show pictures of laptop video card AND a motherboard i have fixed by doing so. will it always work? of course not. even taking a small number of those incredibly tiny fuses and getting them reset can be daunting. and as stated above it very well could be something you cannot locate with the naked eye/multimeter but just because its hard doesn't mean you are automatically out 200 or more. at least not imo and i for one like to do everything in my power to bring something back from the dead before having to pay for transplants again, if you have a giant folding paperweight otherwise, and dont currently have the cash to replace whatever. what have you got to lose? youd be surprised at what can be done by hand, even though tedious, inefficient and often times a temp fix. just my 2c. has saved me a few hundo to say the least. but as well as all of that i will say do not get your hopes up while you sort through the obvious (burnt/visible fixes) issues and either give up or further dmg through frustration when a single minor repair didn't restore everything. just remember that if you are going to take the long road, dont expect it not to be a project/something you can take care of instantly, especially if you are new to that kind of tinkering. but the internet is at your fingertips, every fuse, every circuit/motherboard has specs/blueprints/maps out there, more info than you can fathom and plenty for you to get your hands on/ use to your advantage. just remember that it doesn't need to look like a machine did it in order to function. that realization has allowed for some legendary jimmy rigged computers around these parts. one of our laptops is now housed in a plywood/cardboard/duct tape case because it will no longer fit snug and pretty in its stock home. yet its still one of the higher end pcs and has lasted much longer than it would have otherwise. again, try if nothing else. i put many holes in many electronics with solder guns/irons before i started repairing them. and dont forget electronics are pretty universal when you get down to that level, dont be afraid of cracking open old things to get the fuses and caps or other components you may not have lying around otherwise. just make sure you are replacing them correctly. google is your friends, if you want to save money you will be reading A LOT. consider it a new hobby