Dell Inspiron 1720 Won't Power Up

I’ve got a Dell Inspiron 1720. It’s a few years old and out of warranty.

Last night it developed a problem. Pushing the power button does nothing. The computer won’t start.

But. . . this is one of those Dells that came with that (useless, to me) Dell Media Direct thing. There’s a separate power button for that, and on pushing that button, the computer starts up the Dell Media Direct program. So I don’t think there’s a power problem here.

Any thoughts? Any help would be much appreciated.