How do we find the normal modes and resonant frequencies without making a clever guess? Well, you can get a more complete explanation in an upper-level mechanics course, but the gist of the trick involves a little linear algebra. The idea is the same for any number of coupled oscillators, but let us stick to our example of two oscillators.
normal mode, linear algebra, chapter 3, three
Physical Sciences and Mathematics | Physics
Torre, Charles G., "03 How To Find Normal Modes" (2014). Foundations of Wave Phenomena. 20.