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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...

Narasimham
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  • how would we use them to insight into CN's? – JMP May 01 '15 at 09:59
  • The sum and product are much simpler than the difference and quotient. Go for the simple, every time. The discriminant is important, the difference is very much like the discriminant, only more complicated, so not likely to give any insight we don't already get from the discriminant. – Gerry Myerson May 01 '15 at 10:01
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    'In school we are taught the sum and product of roots of y=ax 2 +bx+c .' so if $(x-r_1)(x-r_2)=0$, you are taught how to add $r_1$ and $r_2$, and also how to multiply them? That's pretty cooool! – JMP May 01 '15 at 10:01
  • I'm tempted to answer: 1) No 2) No – JMP May 01 '15 at 10:04
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    The reason the sum and product are great, but the difference and quotient are less great, is that the sum and product only depend on the (multi)set of roots, whereas in order to describe the difference and the quotient you need to pick a "first" root and a "second" root; that is, you need to break the (Galois) symmetry between the roots. – Qiaochu Yuan May 01 '15 at 10:17

1 Answers1

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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.

Bernard
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