# Quadratic Equations:

# A Different Look

### By: Kimberly Young

## We often look at the graph of
quadratics. We are very interested in seeing how a,b, and c affect
the graph and the solutions. The solutions, or roots, are the
emphasis of this investigation.

## Consider the equation . How do we
expect b to affect the roots of this equation? Let's consider
this graph on the xb-plane.

##

## We can consider a specific b-value
and determine from the above graph if it will have a solution
and/or what the specific value(s) of the solution will be. Consider
b=4.

## Notice that the line b=4 creates
a parallel line to the x-axis and the intersections point of the
line and the curve correspond to the solutions of the original
equation when b=4. Also, from looking at the graph on the xb-plane,
we see that the original equation has no solution when -2<b<2.

# What happens to the graph
on the xb-plane as the value of c changes?

#

## Consider the case when c=-1 rather
than +1. This yields the following graph. The blue graph is when
c=+1 and the red graph is when c=-1.

## When c=-1, we can see that when
b is a particular value, the original equation will have two real
roots, one positive and one negative root.

## This brings up the question,
for what values of c will I always have two solutions?

## Consider the following animation.
The red line is a graph of the c-values when -10<c<10.

## From this animation we can see
that different c-values affect the number of solutions for all
b. If c<0, there will always be two real roots, one positive
and one negative. When c=0, there will be only one real root.
When c>0, the original equation will either have two real roots,
one, or none.

##

## We know from our studies of quadratic
equations the discriminante of the quadratic formula determines
the number of roots a quadratic equation has. Now the following
equation corresponds following discriminant . When the discriminant
is negative, there are no real roots. So when there will be
no real solution. We can conclude:

## -When , the equation
has two negative real roots.

## -When, the equation
has one negative real root.

## -When , the equation
has no real roots.

## -When , the equation
has one positive real root.

## -When , the equation
has two positive real roots.