How many speakers would you need for a quadraphonic sound system?

In theory, four. 'Quadro' means 'four'. It's a four-way system.

In practice, anything between four and six. Designs with a center and a subwoofer are sometimes called 'quadrophonic', because they don't truly receive a 5.1 signal, but rather - derive the low frequencies using band-pass filters, and get the 'center' signal from somewhat-merging the front-left and front-right together.

Do keep in mind that anything driven from a stereo output is stereo, no matter how creative the hardware designers get. Pay attention to the number of jacks you get with a particular sound system - a 5.1 system will have three or four (good quality systems will also sport an S/PDIF connector, through which a 5.1 audio signal encoded using Dolby DTS or some other system can be sent), and a quadro system should have two stereo jacks.

Also, remember that in order to get more than two speakers, your amplifier/primary stage/device/computer must also feature more than one output for this purpose. If this is a PC, the software used for playback must also support these features. Regardless of the device, the source (DVD, music file etc.) must also provide more than two signals encoded on them, in order to enable quadro, 5.1 or 7.1 speaker configurations. If the DVD you're playing has only stereo sound, you'll only get stereo.