2 Parabolas Intersecting at 2 Know Points.

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It is impossible to determine unique equations for two parabolas intersecting at two known points, specifically (0, 0) and (50, 3.44). An infinite number of parabolas can be constructed that pass through these points. These parabolas can be expressed in the form y = ax² + bx, where c equals zero due to the point (0, 0). By substituting the second point into the equation, the relationship between coefficients a and b can be established, confirming the infinite solutions.

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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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