Toshiba Satellite L775 Starsts But Screen Stays Black

My suggestion would be to unplug the cord from the connection on the computer, then remove the battery. After you've done that, press and hold the On/Off switch for about 60 seconds, then, after you've done that connect everything back together, and see if it will reboot. If not, check the hard drive, and the ram and make sure they are all connected tightly.

 The "black screen of death" is usually very easy to repair on a laptop. First turn your computer off by pressing and holding down the start button. Next unplug the computer and then remove the battery. Press and hold down the start button for about 60 seconds. Release the button and plug in the computer. Press the start button and the display should return. Replace the battery. If this does not work, try it again up to four times. No results usually indicates a bad mother board. Hope this helps.

But according to your description, the most possible cause is that your laptop charger is malfunction. So try a new charger to verify the real issue. After all a laptop charger is much cheaper than replacing the motherboard. I recommend one for you:

Replacement Toshiba Satellite L775 AC Adapter 120Watt 19V 6.3A