Harristotelian Logic

Let

z = (x-7)(x+7)

and

x^2 - 6x + 35 = 0, so

z = x^2 - 49, so trivially I have x = sqrt(z + 49) and can now substitute out x, to get

z + 49 - 6sqrt(z + 49) + 35 = 0, so

z + 84 = 6sqrt(z + 49)

and squaring, gives

z^2 + 168z + 7056 = 36z + 1764

so

z^2 + 132z + 5292 = 0.

And you know that each solution for z must share factors in common with 7, with x, as that's the entire point of the construction z = (x-7)(x+7).