We start with what looks like a fairly simple equation
and its graph below.
Now consider the equation above when you graph it in the xb-plane. So we are going to look at the roots of the original equation for different b-coefficient values.
Let's say we want to determine the roots when b=3. In other words, what are the roots when x^2 +3x +1=0.
When the b-coefficient is 3 for this equation, it has two roots, approximately x = -.382 and x = -2.618.
Let's take a look at what happens when c = -1 rather than +1.
It becomes apparent from this graph that in order for the equation x^2 + bx + c to have roots when the value of the b-coefficient is between 2 and -2, the value of c must be between 0 and -1. You can tell from this graph by the way the parabola approaches the origin and finally breaks.