1. Visual Algebra for Teachers
Activity Set 3.5
CLASS PPTX

2. REAL NUMBERS AND QUADRATIC FUNCTIONS
Visual Algebra for Teachers
Chapter 3
REAL NUMBERS AND QUADRATIC FUNCTIONS

3. the Quadratic Formula and
Visual Algebra for Teachers
Activity Set 3.5
Completing the Square,
the Quadratic Formula and
Quadratic Graphs

4. MATERIALS
Black and red tiles, white and opposite white n-strips and black and red x-squares
Graphing calculator with table functions (recommended)

5. PREP Review Algebra Piece Solution for x2 = 4
The dimensions of the black x-square must be .

6. Review: Solution for x2 = 2
According to the algebra piece model, the dimensions of the black x-square must be
To solve:
we “take the square root of both sides” and keep in mind that we should determine both the positive and the negative answer.

7. Taking the square root Suppose we wish to solve
Using our previous technique, “take the square root of both sides” yields:
(it is redundant to write ± on both sides; why?).
This simplifies to

8. The Quadratic Formula
You may have noticed that all of the quadratic functions in Activity Sets 3.2 and 3.3 factored and that all of the corresponding x-intercepts and intersection points had integers or simple fraction x-values.
Of course there are many quadratic functions where x-intercept or intersection x-values are not integers or simple fractions.
For equations such as
we can find the x-intercepts by solving the equations such as , but to solve quadratic equations such as:
we need a more powerful technique.

9. The Quadratic Formula We will start with a general quadratic equation
We will manipulative the equation until we create a square left side set equal to a positive number right side.
After taking square roots, we will have derived a formula (the Quadratic Formula) we can use to solve any quadratic equation.
The technique we will use for creating a square left side is called: Completing the Square.

10. Completing the Square The values of a, b and c are not fixed.
The sizes and colors in the following diagrams are just for illustration.

11. Completing the Square
This is what we wish to solve:

12. Completing the Square
This is simpler if there is only one black x-square, so divide everything by a (remember a  0). Unlike the rest of the “just for illustration” components of the diagram; the remaining x-square will actually be one black square.

13. Completing the Square
To facilitate making a square left side and a number right side; move c/a to the right side.

14. Completing the Square Our goal is to make a square on the left.
If we ignore the lack of black or red tiles on the left for the moment, we can divide the b/a x-strips into two equal piles, each of size b/(2a) and form the following partial square on the left side of the equation.

15. Completing the Square
We can see that in order to Complete the Square (fill in the hole), we must add
black tiles to both sides.
“hole” dimensions are

16. Completing the Square
Add black tiles to both sides

17. Completing the Square
The symbolic formula looks complicated, but we can use the edge set dimensions to factor the left side of the symbolic equation.
We can also add the fractional right hand side
(see the next slide)

18. Completing the Square

19. Completing the Square
Since the left side of our equation is a square and the right side of our equation is a positive number, we can “take the square root of both sides” and see the following:

20. The Quadratic Formula
We solve for x
and we find:

21. Quadratic Formula Example
Suppose we wish to find the x-intercepts for:
First we set the function equal to zero and then we can use the quadratic formula.
Note: a = 1, b = 2 and c = -1

22. Quadratic Formula Example
Using the quadratic formula:
The two x-intercepts are (approximately):
(-2.4, 0) and (.4, 0)

23. and with CTS Notice if we complete the square directly for
we would have:
Note specifically that you don’t have to figure out the factors; by CTS you assure the LHS is (x + b/2a)2

24. The graph of y = x2 + 2x - 1

25. Question #1a
Does the quadratic formula always work and does it always give two real number solutions?
Analyze the function and find each of the following (if they exist). Graph the function, label all of the key points.
y-intercept
x-intercepts
Turning Point
Range
Factored Form

26. Question #1b
Use the quadratic formula to find the x-intercepts for the quadratic function (note b = 0).
What happens?
How does this relate to the graph of the function?

27. Question #1c
Does the quadratic formula always work and does it always give two real number solutions?
Analyze the function and find each of the following (if they exist). Graph the function, label all of the key points.
y-intercept
x-intercepts
Turning Point
Range
Factored Form

28. Question #1d
Use the quadratic formula to find the x-intercepts for the quadratic function.
What happens?
How does this relate to the graph of the function?

29. Question #1e
Does the quadratic formula always work and does it always give two real number solutions?
Analyze the function and find each of the following (if they exist). Graph the function, label all of the key points.
y-intercept
x-intercepts
Turning Point
Range
Factored Form

30. Question #1f
Use the quadratic formula to find the x-intercepts for the quadratic function.
What happens?
How does this relate to the graph of the function?

31. Question #2 (class discussion)
It appears from the previous activity, that a quadratic function can have:
Zero x-intercepts
One x-intercept or
Two x-intercepts.
How can you use the components of the quadratic formula to see this before working out the entire formula?

32. Classwork (in teams, as assigned) 3.5 i) and ii)#3 #4 #5 #6 #7 #8 #9
#10

33. Homework: Monday, 11/10 See HW schedule 3.5 i)
HW Quiz (3.5i) due before 2 p.m. next class
Review 3.5_class PPT, see Moodle for quiz
Prep PPT
No new
Turn in hw (3.5i) due 2 p.m. in one week
See HW schedule

34. Homework: Wednesday, 11/12 See HW schedule 3.5 ii)
HW Quiz (3.5ii) due before 2 p.m. next class
Review 3.5 Activity Set, see Moodle for quiz
Prep PPT/Quiz (3.6) due before 2 p.m. next class.
See Moodle for posted PPTX (to read) and quiz
Turn in hw (3.5ii) due 2 p.m. in one week
See HW schedule