Is my graphics card broken?

hi guys, ive inherited this computer off my brother, and it should be a lot better than it is, cause im getting random blue or red pixels at random points, my games run awfully, and in-game, i very often get these strange graphical anomalies such as incorrect textures or 'beams' of random colour....

Get Slim Cleanse

It could also be your LCD or wrong drivers.

Full system specs please.

The first thing to do is to open the rig up and blow all the dust out of it. Dust and debris in the various heatsinks and fans, can cause overheating.
While you're at it, make sure that all the fans turn freely and run when the machine is powered up.
And also, while you're doing that, remove the video card and re-insert it. Often, theses sorts of problems are due to the video card shifting in it's socket while the computer is moved around - especially if it had a ride in a car.

Beyond that, we'd need more info. But, if it's not overheating, it could be bad RAM on the video card.