X-axis coordinates of outer control points (only) for a Quadratic Bézier curve through 3 points

X-axis coordinates of outer control points (only) for a Quadratic Bézier
curve through 3 points

I am interested in the distance between the 2 outer (left & right: P0 &
P2) control-points of a quadratic Bézier curve that goes through 3 data
points. The curve's non-equidistant control points are different from the
3 data points (sampled concurrent x&y values from the curve): image.
Is there a shortcut to calculate the x-axis coordinates of these outer
control points (P0 & P2) for my curve? I could then simply fit a parabola
to my 3 points (ax^2+bx+c) and "cut" it off vertically at these 2 points
on the X-axis, so I have a curve with the same shape as my old Bézier
curve, but this time in a simple non-parametric/explicit y=f(x) form!
Or is there another way to calculate this distance?
Hardly an original idea I guess, but any help is mucho appreciado!!