In school we are taught the sum and product of roots of $y= ax^2+bx+c$. But are not the difference and quotient of roots equally important?
$$x_{1}-x_{2} = \dfrac{\sqrt{b^2-4ac}}{a}$$
$$ \frac{x_{1}}{x_{2}} = \dfrac{b(b+\sqrt{b^2-4 ac})}{2 ac} -1$$
Does it not give some more insight into complex numbers?
At school we are told that when discriminant is negative the roots are complex, complicated, the region beyond is for time being forbidden territory and discomfiture is necessary when the number is not real…
EDIT 1 & 2
I wrote this sometime back while seeking loci of constant for segments from a common pole and starting point for $ a r^2 + b r + c =0 $
$$ r_1 + r_2 ; \,r_1 r_2; \, r_1 – r_2; \, r_1 / r_2 ; \ldots $$
to compute and plot using differential geometry. The second one is obvious, a circle. others are not so obvious.
EDIT 3:
To derive $ r_1 * r_2 $ as constant another condition needs to be incorporated to get to the circle, however…
Best Answer
The main reason is Newton's theorem on symmetric functions:
A rational symmetric function of two variables $f(x,y)$ (i .e. a rational function such that $f(x,y)=f(y,x)$ for all $x,y$) can be expressed as a rational function of $s=x+y$ and $p=xy$.
That's why $s$ and $p$ are called the elementary symmetric functions. They generalise to functions of more than two variables. For instance, the elementary symmetric functions of $3$ variables are $x+y+z$, $\,xy+yz+zx\,$ and $\,xyz$.
Difference and quotient are not symmetric functions.