This problem will focus on the exploration of the XB plane.
We will start by graphing 
in the XB plane:
Since we are in the XB plane, this means that the x-axis is still the horizontal axis, but the y-axis is now represented with the values of b. Thus the values of b are on the vertical axis.
Through the graph, we can see that there appears to be two asymptotes. These are at x=0 and b=-x.
We will now try to find the number of real solutions for b. We can do this graphically by drawing a horizontal line for some value of b and see where it intersects the drawing. Below is an animation when this line varies from b=6 to b=-6.
We can see that when -2>b>2 there are two solutions, when b=±2 there is one solution, and there are no solutions when -2<b<2.
Now we can examine when there is a real solution for the general case. First, we will solve the discriminate for b:
Then we can use this value for b in the general quadratic equation to see:
This means when 
, there is only one solution to the equation.
We can now examine the equation: 
.
I graphed this equation and then varied the values of c to get the following results:
Remember the real roots of the equation can be found by setting b equal to a value, so I have varied b from -6 to 6 in the image below. This should make it clearer as to how many roots each graph has:
Thus a pattern can be seen for varying values of c:
When c>0, there can be two, one, or no roots of the equaiton
When c=0, there will always be two roots (one of which is x=0)
When c<0, there will always be two roots (one will be positive and the other negative)