2019-10-25, 09:57   #2
Nick

Dec 2012
The Netherlands

2·3·271 Posts

Quote:
 Originally Posted by wildrabbitt Does anyone know if the reason quadratics are studied in this way is anything more than just because they can be? Does getting to understand how to solve such questions give a student any skills for things that could be studied later on in maths? I'm wondering if this has a connection with group theory but I don't know what the connection is, if there is one.
In this case, the coefficients of the quadratic equation are all integers but the roots α and β are irrational numbers.
However, since $$(X-\alpha)(X-\beta)=X^2-(\alpha+\beta)X+\alpha\beta$$, the values α+β and αβ are integers too.
What's more, it follows that any symmetric expression in α and β (i.e. which stays the same when you permute them)
can be written in terms of α+β and αβ and therefore is also an integer.
So this is a way of staying inside the integers instead of having to calculate approximately with irrational numbers, which makes it of great practical importance.

You are right that there is also a connection with group theory.
Permuting the roots of a polynomial equation gives us important symmetries
(known as automorphisms) n the fields containing them, which help us understand their structure.
This was first worked out by Évariste Galois, a French mathematician who died aged 21 in a duel, and the theory is still known today as Galois Theory.
It's quite a showpiece of mathematics!