The graph of f(x) = x^2 - 4x + 5 below in the xy-plane opens upward and has a vertex above the x-axis. Therefore, it has no real x-intercepts and the solutions to the equation when the function is set equal to zero turn out to be complex numbers. If you were to stretch out the x-axis into the complex number plane, you get the three dimensional graph shown here. Below the vertex, there is another parabola. It is in a plane that is perpendicular to the real x-axis. The two complex roots are the intercepts where this second parabola intersects the complex x-plane.

Problem Set
1) Set f(x) equal to zero and solve to demonstrate that the roots really are 2+i and 2-i.
2) Show that f(2+i) really does equal zero.
3) Show that f(2+3i) is a real number.
4) Show that f(1+i) is not a real number.
5) Make a conjecture about the values of a and b for which f(a+bi) is a real number. Explain how you arrived at your conjecture.
6) Prove that your conjecture from above is true.