Let us define partial functions
$$f_+,f_- : \mathbb{R} \leftarrow \mathbb{R} \times \mathbb{R} \times \mathbb{R}$$
so as to return the zeros of the quadratic $ax^2+bx+c$ whenever they exist, such that if $a > 0$, then $f_+(a,b,c)$ is the larger of the two roots, and $f_-(a,b,c)$ is the smaller of the two.
In particular, we define:
$$f_+(a,b,c) = \frac{-b + \sqrt{b^2-4ac}}{2a}, \qquad f_-(a,b,c) = \frac{-b - \sqrt{b^2-4ac}}{2a}$$
So $f_+(a,b,c)$ and $f_-(a,b,c)$ are proper iff $b^2 \geq 4ac$ and $a \neq 0$.
For example, $f_+(1,-1,-1)$ is the golden ratio.
Question. Apart from the obvious stuff, like
- $(2af_+(a,b,c)+b)^2 = b^2-4ac$
- $(2af_-(a,b,c)+b)^2 = -(b^2-4ac)$
- $a(f_+(a,b,c)+f_-(a,b,c))=-b$
whenever both sides of the equation are proper, do $f_+$ and $f_-$ satisfy any other interesting identities and/or relationships to each other and/or to addition and multiplication? For example, can we say anything interesting about $f_+(a+a',b,c)$ or $f_+(aa',b,c)$ or $f_+(f_+(a,b,c),d,e)$, etc?
Furthermore, does $\mathbb{R}$ equipped with these partial functions and possibly one or two others form an interesting partial algebraic structure in its own right, and live naturally in a well-behaved category of similar such structures?
With Viete's Formulas and some simple algebra, we see that $cx^2 + bx + a = 0$ has roots that are inverses of the roots of $ax^2 + bx + c = 0$, so
$$\left\{f_-(c, b, a), f_+(c, b, a)\right\} = \left\{\frac{1}{f_-(a, b, c)}, \frac{1}{f_+(a, b, c)}\right\}$$
It should be possible to generate a large amount of such relations using similar processes (I'm sure textbooks pertaining to quadratic equations have many of such root-transforming problems).
Another example I've worked out:
By finding an equation whose roots are the squares of the roots of $ax^2 + bx + c = 0$,
$$\left\{f_{\{-,+\}}(a^2, 2ac - b^2, c^2)\right\} = \left\{\left(f_{\{-, +\}}(a, b, c)\right)^2\right\}$$