## Rekha Payor

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)

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