A region R is bounded by the line y = x +2 and a parabola with vertex at (0, 0). Find the equation of the parabola such that the numerical value of the area of region R is the same as the numerical value of the distance traveled along the parabola between the points of intersection with the given line. Final answer(s) should be exact values (where possible) as well as approximations rounded to 7 decimal places.

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