Hee Jung Kim
We will use y instead of using b for this investigation.
I. We will consider the equation , and graph this relation in the xy plane.
The graph of shows a straight line passing through the origin and the y-axis, and the graph of is a hyperbola with two asymtotes x = 0 and y = - x , which can be represented by .
When n > 0, we get
Therefore, the graph of always intersects the horizontal line for any real number b at two points whose x-coordinates are negative and positive respectively. This implies that for any real number b, n > 0 and n = 0 has always two real roots. In particular, one of two roots is positive, and the other is negative.This is a nice approach to investigate a pattern of roots of the quadratic equation .
II. How about a pattern of roots of for n < 0? Again, by replacing b by y, we will consider the equation for n < 0, equivalently for n > 0, and its graph in the xy-palne.
We note that even though and , n > 0 have the same asymptotes x = 0 and y = - x or , their graphs are different hyperbolas. In this case, setting a constant for n, say n = 1, let's vary y from -5 to 5. Then we have the following animation.
The graph of a system of two equations can be interpreted in relation of the solution pattern of for different b as follows.
Case 1. has two negative real roots if b > 2 because there are two points of intersection whose x-values are negative.
Case 2. has one (repeated) negative real root if b = 2 because there is only one point of intersection whose x-value is negative. The graph approximately shows the root x = -1.
Case 3. has no real roots (in other words, two complex roots) for -2 < b < 2 because two graphs do not meet.
Case 4. has one positive root if b = -2 because there is only one point of intersection whose x-value is negative. The graph approximately shows the root x = 1.
Case 5. has two positive real roots if b > 2 because there are two points of intersection whose x-values are positive.