`by `

`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

`
`

`
`

and

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.