Trying to explain to a kid why quadratic equations matter…
It’s all about relationships. When you look at things simply, you can see that for example, as you eat food, you put on weight. One assumes a relationship, and on a very crude level, one can approximate that, over some small enough interval of time, the weight will correlate with the amount of food linearly. y=ax+b. There is no reason for this to be true. Laws of conservation of mass aside, one does after all produce volatile substances with the food through metabolism, some of which CO2 weigh more than O2. The linear approximation is just the simplest expression of relationship, so we use it.
Real relationships are more complicated usually. So, for example, position of things with respect to gravity, is usually 2nd order (involving x squared) because it is an acceleration, that is a change involving both the dimensions of space and time. Also, areas of things like circles generally involve dimensions-squared. So there are cases, where the solution of a second order equation (quadratic) is the exact solution to the physical problem.
Most of life can be approximated by a Taylor series over a given interval, and finding the coefficients of the terms becomes a useful approximation for the very general problem one is trying to solve over that range. If you make the problem small enough, it is very often approximable by a 2nd order polynomial (basically a harmonic oscillator with a single restoring force to equilibrium). Exact solutions are always available (sometimes imaginary) for polynomials of degree 4 or less. After that, if I remember correctly, the solutions (roots) may not always be expressed in terms of the coefficients, although they can always be found using Newton-Raphson approximations. So, although constructive elegance cannot be found, it is claimed that one can clumsily batter the problem to death to get the needed solutions.
On a more elegant note, I think that Galois showed that there are some group symmetry arguments in the coefficients of a polynomial that must be satisfied in order for the polynomial to be “factorable”. It may be (kind of winging it here), that much as the imaginary number construct provided a dimension to allow unsolvable quadratics to be solved (and eventually returned to nonimaginary space to provide real solutions to problems), that a similar transformation can be found to solve higher order polynomials so that they do in fact radicalize. A heavy group theory problem. Probably worth a Nobel Peace Prize, if they had one in math.
Maybe the future of math…but I don’t know enough to be useful here.