# Elliptic and Parabolic Curves

Algebra Level 4

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

When

$y^2 = -x^3 + x^2 - x + 1 \text{ and } x^2 = - y^3 + y^2- y + 1$

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

Find $f(x)$ in terms of $x$ and $a$.

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