Assignment 3: Quadratic
Equations by Dorothy Evans

LetŐs examine the equation _{} and look at its
roots.

First we will graph the equation as _{} and examine its
roots (i.e. where y=0)

LetŐs see what happens as we change b.

It seems clear that for b=1 there is only 1 root at (0,0)

What about for where b=2, it looks like there are two
roots one at (0,0) and one at (-1,0). Can we prove it?

LetŐs look for a minute where the graph came from

_{}

What would happen if we factored the equation?

Next we would get _{}

Then we utilize our product property to get either

_{} or _{}

Further factoring we know that we can factor_{}again to get

_{} so

_{} is a root.

What about our next equation where b=3?

This one doesnŐt look so simple. ItŐs somewhere between -3 and -2 but how
can we find out where.

LetŐs see what we can do to solve it.

Starting with _{}set y=0

We again factor to get _{}

Then we know x=0, but how do we factor
_{}?

We could do 2 different things, first we could
complete the square to solve or we could use the quadratic formula.

Utilizing the quadratic formula we get
_{}.

Does this answer seem reasonable? We can estimate and get x=-.38 and
x=-2.62.

So what would the general equation be for the roots?

It would appear we could plug in b for b and solve.

So _{}

LetŐs factor _{}

Next we get x=0 and x=_{}

So does this tell us anything about the local maximum
we see on the left side of the graph.

Can we now hypothesize about the value of that local
maximum?

What would you guess?