quadratics
Quadratics
In the form of \[ax^{2}+bx+c\] or \[x^{2}+\frac{b}{a}x+\frac{c}{a}\].
Let \[\alpha\] and \[\beta\] be the roots of the quadratic equation, then \[(x-\alpha)(x-\beta)=0\]. If we expand that equation, we would get \[x^{2}-(\alpha+\beta)x+\alpha\beta\].
By comparing \[x^{2}+\frac{b}{a}x+\frac{c}{a}\] and \[x^{2}-(\alpha+\beta)x+\alpha\beta\], we would get \[\alpha+\beta=-\frac{b}{a}=S_{1}\] and \[\alpha\beta=\frac{c}{a}\].
\[S\] here would refer to \[S_{n}=a^{n}+b^{n}\]. (This applies to both cubics and quartics)
Sums
The sum of roots can be written as \[\Sigma\alpha=\alpha+\beta\], the product as \[\Sigma\alpha\beta=\alpha\beta\] and \[\frac{1}{\alpha}+\frac{1}{\beta}\] as \[\Sigma\frac{1}{\alpha}\] .
An example would be: \[\alpha^{2}+\beta^{2}=(\alpha+\beta)^{2}-2\alpha\beta=\Sigma\alpha^{2}-2\Sigma\alpha\beta\] and \[\frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}}=\Sigma\frac{1}{\alpha^{2}}\].