**BY: ERICA FLETCHER**

**Graphs
in the xb plane.**** **

**Consider
the**** basic quadratic equation**

**where**** b
is any real number.**

**Now
graph this relation in the xb plane. We get the
following graph. The graph below shows how the values of x and b relate to each
other. The x values are along the x-axis and the b values are along the y-axis.**

**If
we take any particular value of b, say b = 5, and overlay this equation on the
graph we add a line parallel to the x-axis. If it intersects the curve in the xb plane the intersection points correspond to the roots of
the original equation for that value of b. Therefore, we have the following
graph.**

**We
notice below, for each value of b we
select, we get a horizontal line. **

**Notice
the gap between the two sections of the graph. Between -2 < b < 2, there
are no values for this graph. It is very clear on a single graph that we get
two negative real roots of the original equation when b > 2, no real roots
for -2 < b < 2, one positive
real root when b = -2, and two positive real roots when b < -2. **

**Lastly, let's consider the case when c = -1 rather that +1.**

Consider the following equation again:

where a = 1, b =1, and c = -1.

If we look at the discriminant now we see that the above graph makes sense because we will always get two roots.

Let's look at the discriminant:

Since c = -1 the discriminant will always be greater than zero.

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