# Motivating algebra from quadratic equations

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.

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Bennet: Very nice question. At an even more basic level, there is some very nice mathematics that comes from solving linear equations. –  André Nicolas Jul 3 '11 at 20:03
Is "motivate based on the quadratic formula" meant to subsume the study of rings constructed by arbitary (iterated) quadratic extensions of rings or domains, starting from some class of "foundational" rings? –  Bill Dubuque Jul 3 '11 at 20:56
@ Bill: I'm thinking of what I can indicate/motivate for young people still at school, who will not know much algebra but will know quadratics. "There's a whole world out there to explore ... " So "this is a beginning and you can go further than this" (iteration) is within the scope. –  Mark Bennet Jul 3 '11 at 21:16
So thinking further about Bill's comment we could do constructible polygons. –  Mark Bennet Jul 4 '11 at 9:38

You can motivate a lot of Number Theory by picking a positive integer $n$ and asking, what if, instead of requiring $ax^2+bx+c$ to be zero, we settle for it being a multiple of $n$? From this you can define congruences, you can see what makes primes so special by reducing the general case to the case where $n$ is prime, you can introduce finite fields, you can motivate quadratic reciprocity (which is the question, if $p$ and $q$ are odd primes, and there is an integer $x$ such that $x^2-p$ is a multiple of $q$, is there an integer $y$ such that $y^2-q$ is a multiple of $p$?), and from there the sky's the limit.
Yes, see Cox's Primes of the form $x^2 + ny^2$. –  lhf Jul 4 '11 at 0:30