Lessons from Perfect Square

Two lessons:

$1.$ Binary search works on any monotonic function, not just sorted arrays. Here, $x^2$ increases as $x$ increases, so you can binary search over $x$

$2.$ When forbidden from using built-in functions, binary search often replaces them efficiently

This same pattern applies to cube roots, nth roots, and other inverse operations. If you can check a candidate answer quickly, binary search finds it in log time.