The document provides step-by-step instructions for solving 4 math problems: 1) Finding the maximum water volume of a plastic water bottle given the plastic area. 2) Isolating variables, long division, and factoring to simplify a quadratic equation in order to graph it and find the domain. 3) Factoring and solving for zero to find the values of x that satisfy a polynomial equation. 4) The steps for completing the square to derive the quadratic formula.

2. 1st Problem
If you have 500cm2 of plastic, and have to make
a water bottle from it. What is the maximum
amount of water it can hold?
-Find the dimensions of this water bottle

4. 4) Lets plug our new variable for
“h” into the Volume equation
5) Now that there is only 1
variable in this equation, we can
simply put it in our calculators
and solve for the maximum
Be careful putting this equation in the
calculator.
6) Since this graph is a cubic,
there are 2 maximums. Since
one of the maximums has a
negative x value, it can’t be a
maximum because lengths can’t
be negative.

5. 7) We now have to find the “h”.
Lets use our equation that finds
the “h”.
8) We can easily check our
answers by plugging them into
the S.A. equation and see if we
get 500.
9) Answer the question

6. 2nd Problem
• Sketch the graph and find the
domain
- 1 root is (x+4)

7. 1) Lets first start this problem
by isolating the variables.
Lets move the “y” variable to the left
2) Now we have to somehow
simplify this long equation with
the x’s. We have to do long
division with one of the roots
we got from the question.
3) We still have to simplify that
equation. So lets try using the
factor by grouping now.

8. 4) Now lets move our new terms
in to the original equation now
5) Lets isolate “y”
6) Now we can graph the rough
outline of the equation now
because we know the x-
intercepts. Plug it into the
calculator also to get the outline
of the graph also
7) Find the domain by looking at
the x-intercepts and the graph on
the calculator.

9. 3rd Problem

10. 1) First we have to know what
this equation is asking.
2) Then, lets get these equations
on one side. Lets subtract the
equation to get them all on
one side
3) Now we should combine the
terms to see if we see any
pattern
We factored out the 4 and found easy factors

11. 4) Now that we got an easy
equation to solve. Solve it for
zero.
5) So in the end, if we plug
these numbers into our original
equation, we equal them out.
We can simply check our answers by doing this,
and if they don’t equal each other, the x’s are wrong

12. 4th Problem

13. 1) The first thing we do
when we complete the
square, is subtract the
“c” value.
2) The next step is to simply
factor out the “a” value.
So we divide “a” by
everything since there is
addition.
3) Then, we have to take
the new b value and have
to cut it in half, then
square it.

14. 4) Then we are going to add
that term to both sides of the
equation.
5) Next, we can make a square
on the right side now.
We do this by cutting the whole b value in half
6)Now lets combine the right
side to make it look nicer
We have to make common
denominators before we add

16. 9) We have solved for “x” in this
equation. Which is known as the
Quadratic Formula.