Quadraphonic L P played with a 2 chan. cart.

Yes.
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One kind of quad LP (SQ encoded) is simply a matrix system where the 2-channel signals are processed to derive four signals. No problem for the cartridge. It will probably sound OK.
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Another kind of quad LP (which I never tried) requires a cartridge with HF response up to at least 40 KHz, because the signal that separates out the rear channels is carried by FM modulation of a carrier in the range of 20 to 40 KHz. (Note that the cartridge does not need to have flat or low distortion in this upper range because the modulation is FM). If you play such a record with a cartridge lacking HF response you might damage the grooves. However, the good news is that because of this quad system the HF response of phono cartridges made a great improvement, and most if not all cartridges sold today would work fine. The quad system died, but we still benefit from it.
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Inpepinnovations...I also used a Lafayette SQ-L decoder, which is now in my collection of defunct audio stuff. It was probably the best of all the logic decoders. Mine is all discrete components and must have cost much more to build than the $99 I paid for it when it was discontinued. My guess is that the circuit was a prototype that they intended to implement in ICs. I modified it a bit and used it as my preamp for several years. It had a bit of gain which made this practical.