Find all values of $k$ so that the graphs of $x^2 + y^2 = 4 + 12x + 6y$ and $x^2 + y^2 = k + 4x + 12y$ intersect.

Quadratic formula ax^2 + bx + c = 0: x = (-b pm sqrt(b^2 - 4ac))/2a

 

This quadratic has a real root when b^2 - 4ac >= 0.

 

x^2 + y^2 = 4 + 12x + 6y

x^2 + y^2 = k + 4x + 12y

 

Graphs intersect when -67 <= k <= 43.