Is it possible to solve this?

Um. You foiled it wrong. It's supposed to be 49x^2 - 42xy + 9y^2.

So:
y = (-42x +/- sqrt((42x)^2 - 4*-9*(13 - 49x^2))) / (2 * -9)
= (-42x +/- sqrt(1764x^2 + 468 - 1764x^2)) / -18
= (-42x +/- sqrt(468)) / -18
= (-42x +/- 6 sqrt(13)) / -18
= (7x +/- sqrt(13)) / 3

... which is the same answer you get from completing the square, if you account for the +/-:

(7x - 3y)^2 = 13
7x - 3y = +/-sqrt(13)
-3y = -7x +/- sqrt(13)
y = (7x +/- sqrt(13)) / 3