This question gave me pause for thought. We have a quadratic equation $ax^2+bx+c=0$. How much algebra can be motivated from the standard solution. Comments point out that the formula does not apply in characteristic 2, and that we need to be able to divide by $a$ and take the square root of the discriminant.
I reckon this gets us thinking about fractions, fraction fields (and even local fields) with the rationals as the field of fractions of the integers (and rational functions from polynomials not far behind).
Then we get quadratic extensions of fields and rings. Including Complex Numbers.
What struck me was that relatively elementary observations could take us a long way. I think of talking to my daughter (age 13) about mathematical ideas and reckon I could do all of the above with her in the Quadratic case.
But the quadratic case has some special features and therefore is not always paradigmatic for general theory.
It has always seemed to me that indicating possible directions of travel in generalising simple results would be of huge benefit in motivating bright youngsters to take up mathematics.
I'm looking for answers which give me insight into how much algebra, algebraic geometry, algebraic number theory I could motivate in an elementary way based on the "formula" for solving a quadratic equation.