The graph of [color=#c51414]f(x) = x^2 - 4x + 5[/color] below in the [i]xy[/i]-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 [color=#1551b5]parabola[/color]. It is in a plane that is perpendicular to the real x-axis. The two complex roots are the intercepts where this second [color=#1551b5]parabola[/color] intersects the complex [i]x[/i]-plane.

Problem Set[br][br]1) Set f(x) equal to zero and solve to demonstrate that the roots really are 2+i and 2-i.[br][br]2) Show that f(2+i) really does equal zero.[br][br]3) Show that f(2+3i) is a real number.[br][br]4) Show that f(1+i) is not a real number.[br][br]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.[br][br]6) Prove that your conjecture from above is true.