LPS & SPS can most definately thrive in the same aquarium. They grow alongside one another on many reefs and both create a calcium based skeleton. Both will thrive with parameters found in natural seawater on the reef, but you will probably need to test for and add these compounds (alkalinity, calcium, magnesium) to the water eventually and keep them stable. And you typically must add more over time as the corals grow out.
As for nutrients like phosphate & nitrates, small amounts are OK and even desirable for both categories. But not enough to promote excessive pest algae species. Very high concentrations of these nutrients over long periods of time can inhibit skeletal growth. Some tanks do well with almost or near zero levels of nit/phos. But levels of ~5ppm/<.02 work better for some, and others can get by with even higher levels.
The main issue with SPS - LPS compatibility is coral warfare. Some LPS corals like bubble, favites & chalices have surprisingly long stinging tentacles and are keen on destroying their neighbors if they can. So attention must be paid to placement & water flow. For a person new to corals, this is the hardest aspect to work out in a mixed reef. You can read up on the topic and avoid obvious mistakes, but almost everyone who keeps stinging corals has to learn at least some of it by trial & error when it comes to placement.