roots of quadratics
Hi,
in the Alevel maths curriculum, there's a topic to do with roots of quadratics.
A typical question would be for example,
The quadratic \(2x^2+3x6\) has roots \(\alpha\) and \(\beta\).
(i) Write down \(\alpha+\beta\) and \(\alpha\beta\).
(ii) Hence show that \(\alpha^3+\beta^3=\frac{135}{8}\).
(iii) Find a quadratic whose roots are \(\alpha+\frac{\alpha}{\beta^2}\) and \(\beta+\frac{\beta}{\alpha^2}\).
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.
