Hooking up powered (active) speakers with non-powered (passive) speakers to a mixer board is actually a common set-up. Often the one set of speakers are set up as the main loudspeakers, while the other set of speakers act as stage monitors so the performers can hear themselves. In order to use passive speakers, you need to use an amplifier, which will push sound out of the speakers. Active speakers have built-in amplifiers.
Connect a 1/4-inch audio cable from the left main out jack on the back or top of the mixer (depending on the model and brand) to the left powered speaker. Repeat for the right main out.
Insert a 1/4-inch audio cable to the left "aux sends" jack on the back or top of the mixer. Connect the other end to the left input on the amplifier. Repeat for the right "aux sends."
Connect a 1/4-inch audio cable from the left out jack of the amplifier to the left speaker. Repeat for the right out of the amplifier.