A PCI-E 3.0 card will work in a 2.0 slot, but yes at half the bandwidth. Granted this still does not matter as GPU's are still are unable to fully take advantage of PCI-E 2.0 bandwidth.
Buddy, whatchu talkin' 'bout?
That Core i3 model is an Ivy Bridge CPU, which has the PCI-e 3.0 controller built in, so you can use the PCI-e 3.0 card as the latest bandwidth IF you have a 3.0 slot. If you put it in a 2.0 slot, you're out of luck.