Its the variables that are unassigned or free, not the formula.
Formula come later. You begin with variables. If all your variables are assigned then your formula cannot be free, if you insist on labeling your formula unassigned or free. The thing is, the terms don't apply to formulae.
What do it means for a variable to be free? It means you haven't decided which natural data it represents. Yes, its as ridiculous as it sounds. Why will you proceed making a formula if you are not even clear about the terms of formula? and How? You shouldn't and you can't.
Do it happen that people make formula with unassigned variables? Yes, a lot. Examples:
- Dark Energy.
The data for how much force of gravity exist in universe do not match with Theory of Relativity. The data suggests more gravity. What scientists then did is put a free-floating variable, labeled it Dark Energy and still have an equation.
The right way to approach is to not propose any equation till you have your act together. What is that Dark Energy? Is it ever observed? Can it be measured independently? The answer is its an imaginary thing nobody ever observed. It means anything the scientist want it to mean, depending on situation.
- Delta as in Delta-X. The Basis of Derivates in Mathematics.
Nobody ever decided what delta is. Its said to be infinitesimally small. Well, if its infinitely small then its zero, isn't it? Its said to be smaller than any number - including fractions - but greater than zero. What?
Its not that I can arbitrary take a value for its length. Say, I decide to measure a thing at intervals of 1000th of its length. No, as per theory of derivates as soon as I decide on any number - 1000th in this case - I am wrong. I cannot put any number on it.
I can think of many more from the top of my head, such as Virtual Particles, Energy Borrowed From Field (once a particle is in an energy pit) etc. But it would become a physics answer then.