(1)                                                            Ax2 + Bx + C = 0,
with non-zero A reduces by dividing through A:  B'=B/A, C'=C/A to:
(2)                                                            x2 + B'x + C' = 0.
This setting:
(3)                                                           -2p = B' = B/A        <=>        p = -B/(2A)   and   q = C'=C/A,
reduces to (4)                                                            x2 - 2px + q = 0.
The function  y = x2 -2px + q is a translation of the standard quadratic:
(5)                                                            y = x2,
since:
(6)                                                            y= x2 - 2px + q       <=>           y-(q-p2) = (x-p)2.
The translation  { y'= y - (q-p2), x' = x - p} transforms     y = x2 - 2px + q    to the standard   y' = x'2 and the x-axis  y=0 to  y' = p2-q. Thus the two roots are found by intersecting the standard parabola   y' = x'2   with the line  y' = p2 - q,   i.e. solving   x'2 =  p2-q  and then pushing the two solutions  (x'1, y'), (x'2, y') back to the x-axis by the inverse translation giving the pair  (p+x'1, 0), (p+x'2, 0).
The point is that we are cutting all the time the same parabola with parallel lines.
By the way notice that points (p,q) for which the corresponding equation touches the x-axis are precisely the points on the standard parabola y=x2.

Leaving the Ax2 + Bx + C = 0 as it is and solving directly with the well known formulas is essentially the same as before except for a similarity by the factor (1/A) by which first we map to a similar parabola (all parabolas are similar).
The dynamic figure changes by moving point (p,q). Points {s1, s2} have a meaning when there are no real roots. They represent then the two complex and conjugate roots of the equation x2 -2px + q = 0.
For general properties of parabolas see Parabola.html . Relating to the standard equation of the parabola see ParabolaParameter.html .