Quasar 32" LED TV. Why did the picture go black with no prior warning?

Check and replace damaged component/s at its back-light inverter board, or replace this board as card basis. If you wish to get some details; check the site linked here. Surf the site with patience. Pull up older posts. It will be best to replace the inverter board as card basis.
http://electronicshelponline.blogspot.com/
In some models of LCD TVS; the backlight inverter and its T'Con board will be a part of the panel itself, and will be a built-in type inside the LCD panel. If so, they can't be replaced. In this case; a panel replacement is the only repair. For some other models, the backlight inverter section circuit will be a part of the main power supply regulator [SMPS]. To such sets, the power supply section circuit too should trouble shoot. Contact any service technician. The circuit will differ with models and versions.
Type in the brand name of your device, in the search box [at the top right] of the display window and the Search will start at once as you type in, and will show you all the related posts. It will make surf easy.