Quadratic equations sometimes have "complex roots":

Complex roots are numbers that include the square root of a negative quantity. To understand this first you need to know about the special number known as i. i is the square root of -1. That's right. You thought there was no such number. How can you have the square root of a negative number? Well you can, and it is known as i which is short for imaginary. You can do arithmetic with i.

A complex number is a number with two parts to it. These parts are called the "real" part and the "imaginary" part.
Examples of complex numbers are: 4 - 6i, 2.1 + 4.7i, 1-i, and so on... In fact any number of the form a + bi, where a and b are real numbers, and i is the square root of -1, is a complex number.

Addition and multiplication of these numbers is achieved using the conventional rules of algebra.

Any two complex numbers such as 4 + 3i and 4 - 3i, which differ only by the sign of their imaginary parts are called complex conjugates.

The product of two complex conjugates is always real. This is easily shown by
(a+bi ) (a-bi) = a²+b².

Because of Properties 4 and 6, we know that both the sum and product of the roots of any quadratic equation are real numbers. In fact it is possible to be clearer than that. If a and b are the roots of ax²+bx+c=0, then:
a+b=-b/a, and a´b=c/a

What is most important about these facts is what they tell us about the midpoint between two points.

Statement 1
The midpoint of the line from A to B has real coordinates when A and B have coordinates that are complex conjugates.
From Property 2 we know that the midpoint, M, of A(a,b) and B(c,d) has coordinates ((a+c)/2, (b+d)/2). Now, if a and c are complex conjugates, then (a+c)/2 will be real (Property 4) and the same goes for (b+d)/2. So the midpoint M will have real coordinates, and can always be plotted - even when A and B cannot!

• there are always two intersection points when two circles are drawn
• when the circles do not touch, these points have complex coordinates and so cannot be displayed
• the midpoint of these two intersections can always be displayed

Now, what about the line passing through the two points of intersection? We must prove a statement involving coordinate geometry and complex number theory.

Statement 2
If A and B are two points with complex conjugate coordinates, then the line joining them is a real line.
Proof
To prove this statement it is sufficient to show two things:
(a) the gradient of the line is a real number
(b) the midpoint between A and B has real coordinates.

First we show (a) is true.
Let A have coordinates (a+bi,c+di) and B have coordinates (a-bi,c-di). By referring to Property 5, you can see that the gradient of AB is 2di/2bi=d/b, which is real.

We proved (b) above in Statement 1.