Is it necessary for students to learn math like the quadratic formula, linear equations, and radicals, assuming they don't go into a profession involving math?
It's part of learning how to think. It's really important.
In school math isn't about math. It's about problem-solving and developing a logical thought process.
General math is good enough until a career path is decided. Once the path is chosen then focus on the skills for that path. If it involves advanced math then learn it, if not then focus on something that you will need to know.