Harristotelian Logic

Here's a simple way to force an unlimited number of quadratics to have roots with the same factors in common with 7.

Start with the basic quadratic

x^2 - 6x + 35 = 0

and now let x = y + 6 + 7k, where k is an integer, and trivially you have that x is coprime to y with respect to 7.

Now then you can substitute out x, to get

(y+6+7k)^2 - 6(y+6+7k) + 35 = 0

so

y^2 + 2y(6+7k) + 36 + 84k + 49k^2 - 6y -36 -42k = 0

so

y^2 + (6y+14k)y + 42k + 49k^2 = 0

and

y = (-(6y+14k)+/-sqrt((6y+14k)^2 - 4(42k + 49k^2)))/2

and for every integer k, you have that y is coprime to x, therefore, for every integer k, you have that the roots must have the same factors in common with 7.