Elliptic and Parabolic Curves

Algebra Level 4

Let f(x)f(x) be a quadratic function.

When

y2=x3+x2x+1 and x2=y3+y2y+1 y^2 = -x^3 + x^2 - x + 1 \text{ and } x^2 = - y^3 + y^2- y + 1

are simultaneously graphed, there are three intersections. f(x)f(x) passes through all three of them, and there is one intersection where x=y=ax=y=a.

Find f(x)f(x) in terms of xx and aa.

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