" Introduction to phantom graphs" , Philip LLOYD ... P.31 - 41 ,
While teaching “Solutions of Quadratics” I was emphasising the idea that, in general, the solutions of equations such as ax² + bx + c = 0 are obviously the points where the graph of y = ax² + bx + c crosses the x axis. I started to be troubled by the special cases of parabolae that do not even cross the x axis. We say that these equations have “complex solutions” but physically, where are these solutions? With a little bit of lateral thinking, I realised that we can physically find the actual positions of the complex solutions of any polynomial equation and indeed many other common functions! The theory also shows clearly and pictorially, why the complex solutions of equations with real coefficients occur in conjugate pairs.
These ideas in this paper are completely Philip LLOYD own original work.