Universal quadratic formula? Is there any way to write the quadratic formula such that it works for $ac= 0$ without having to make it piecewise?
The traditional solution of $x = (-b \pm \sqrt{b^2 - 4ac}) / 2a$ breaks when $a = 0$, and the less-traditional solution of $x = 2c / (-b \pm \sqrt{b^2 - 4ac})$ breaks when $c = 0$... so I'm wondering if there is a formula that works for both cases.
My attempt was to make the formula "symmetric" with respect to $a$ and $c$ by substituting $$x = y \sqrt{c/a}$$ to get $$y^{+1} + y^{-1} = -b/\sqrt{ac} = 2 w$$
whose solution is
$$y = -w \pm \sqrt{w^2 - 4}$$
which is clearly symmetric with respect to $a$ and $c$, but which doesn't really seem to get me anywhere if $ac = 0$.
(If this is impossible, it'd be nice if I could get some kind of theoretical explanation for it instead of a plain "this is not possible".)

Answering my own question, but I just realized this algorithm on Wikipedia works if we cheat a little and don't consider $\operatorname{sgn}(x) = |x| \div x$ a "piecewise" function:
$${\begin{aligned}x_{1}&={\frac {-b-\operatorname{sgn}(b)\,{\sqrt {b^{2}-4ac}}}{2a}}\\x_{2}&={\frac {2c}{-b-\operatorname{sgn}(b)\,{\sqrt {b^{2}-4ac}}}}\end{aligned}}$$
