Warranties for computers cover hardware faults.
If the problem is due to something you have done, knowingly or otherwise - not using a proper antivirus, installing pirated software with nasties, or even just getting a "drive by" malware infection from a compromised web site - then that is not covered by warranty.
If it's an actual hardware fault, you have a warranty claim.
If it's a software problem, even if the machine will not boot at all, it's not covered by warranty.
Your only possible getout with that is to get them to show you how to do a factory reset on it - they should have provided media or files on the machine to do that. Ask them for it if not.
To have a chance of suing them [successfully], you woudl have to get the machine repaired at a different, independent computer shop and get a written report on what the fault was.
If that said it was a hardware problem, then you could sue the original supplier for the cost of the repair that they should have done. If it's a software problem, you are out of luck - it's a PIBCAK...
http://www.urbandictionary.com/define.php?term=pibcak
For info, the standard routine for a new computer warranty claim is they take you through the procedure to do a "factory reset" on the machine, which overwrites everything on it and sets it back to how it was the day it was bought.
If it works then, the problem was caused by something the owner did and is not a fault or covered by warranty.