Finding parabola equations given two intersection points

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deltapapazulu
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Is it possible to find the equations of 2 parabolas intersecting at 2 known points? For example, (0, 0) and (50, 3.44).
 

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No. You can construct an infinite number of parabolas passing through two given points.

Even if you require that the parabolas have vertical axis of symmetry there exist an infinite number of parabolas passing through (0,0) and (50, 3.44):
Any such parabola can be written in the form y= ax2+ bx+ c. Setting x=0, y= 0 give 0= c so we still have any parabola of the form y= ax2+ bx passing through (0,0). Set x= 50, y= 3.44 gives 3.44= 2500a+ 50b so b= 0.0688- 50a. Any parabola of the form y= ax2+ (0.688- 50a)x with a any real number passses through (0,0) and (50, 3.44).
 
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