With the equation $ax^2+bx+c=0$ in mind, I'd like to know if the relations $$ s_a(x)=\frac{-b\pm\sqrt{b^2-4xc}}{2x},\quad s_b(x)=\frac{-x\pm\sqrt{x^2-4ac}}{2a},\quad s_c(x)=\frac{-b\pm\sqrt{b^2-4ax}}{2a} $$ are ever useful and if not, why?
My guess is "no" since they don't seem to be part of any standard mathematics curriculum (that I've encountered). The guess is strengthened by the lack of results I find on Google.
Even then, I feel that it is a natural curiosity for any student learning about the quadratic formula to try and graph the output of the quadratic formula against $a,b$ or $c$. The graphs of each relation reveal some interesting patterns of solutions for quadratic equations depending on how $a,b,c$ change.