No. Academic Magnet is a public school, not a private school but both schools do I believe require academic testing to be accepted. In Greenville county, all schools are open to kids to attend via special permission, application to magnet programs, or payment and testing. I would assume it is similar in the lower state at least with the the governors school and magnet schools. If you find a private high school that recruits and pays the tuition for girls soccer, please let everyone know since there are a lot of good soccer players in the state whose parent's probably wouldn't mind having a free private school education.
