NAME:____________________________
HOUR: _________
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1. What is a quadratic equation(look up on ipad)?
2. What does the graph look like ?
3. Circle the functions that are quadratic ?
a) f (x) = 2x + 5
e) h(x) =
b) g(x) = x 2
1
x2
c) y =
3x − 1
e) y = 9 − x 2
d) y = 3x 2 + 5x − 2
f) y = x 3 + 4x 2 + 6
4. A rocket is shot off ground the ground at time t , the height the rocket is off the ground is
given by the equation h(t) = −16t 2 + 80t
a) Determine the height for each time value in the table
b) graph the time -vs- height
t (Time)
h(t) height
0
1
2
2.5
3
4
5
c) When does the rocket land back on the ground?
d) What is the maximum point? What does this point represent in terms of the rocket?
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PART 1 FACTORING QUADRATICS
Goal of this section: is to write x2 - 9x + 20 in the factored form (x - 5 ) ( x - 4).
We will start by reviewing multiplying quadratics and then proceed to factoring. In unit4, we
multiplied (x + 1)(2x - 3) by using an area model
so
( x + 1 ) ( 2x - 3 ) = 2x2 - 1x - 3
5. Multiply each:
a) 4x(-2x+7)
answer:
b) (3x+6)(2x - 5)
answer after like terms combined:
c) (2x+4) (3x + 5)
d) ( -4x + 3) (2x - 5)
answer:
answer:
e) (x - 6) (x + 5)
answer:
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Factoring is reversing the process of the multiplication.
Example: 15 is 3 * 5 in factored form
Goal of this section: is to write x2 - 9x + 20 in the factored form (x - 5 ) ( x - 4).
The factored form of the expression has many practical applications in mathematics.
6. Determine the expressions that were multiplied if the area model products are given to you
instead.
a)
b)
d)
e) challenge
c)
f) Discuss with your classmates your strategy for finding the factor expressions.
Trinomials with the quadratic term x2
7. Multiply the binomials together, using any method.
a)
( x + 3) ( x + 2)
b) ( x + 4) ( x - 2)
c)
( x - 5) ( x - 4)
d)
( x + 3)2
8. Each answer in question 7 had the term x2 in its product .
Explain where the x2 term came from.
9. If d and e are any number will all products of the form (x + d ) ( x + e) contain the term x2 ?
10. The answer for 7a) is x2 + 5x + 6 . Explain where the 6 came from. Does your
explanation also work for 7b) , 7c) and 7d) ?
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11. If the binomials (
product?
+ 4) (
- 6 ) were multiplied together, what number must be in the
12. We now know when ( x + 3 ) ( x + 5) are multiplied together, two of the terms
in the quadratic will be x2 and 15 .
13. The other terms would be 3x and 5x, so the final product answer would be x2 + 8x + 15
Explain were the 8x came from.
14. Trying to factor x2 + 6x + 8, Lauren placed the x2 and 8 in the area box
but didn’t know what the other two missing terms would be. Help her determine
the missing terms and then find the factors that multiply to be x2 + 6x + 8.
15. Check your factored answer to question 14 by multiplying the factors together. They must
multiply to x2 + 6x + 8
16. To factor x2 + 8x +12 , we know that the x2 and 12 term can be placed in the area box.
Explain what must be true about
a) the sum of the two terms that will go in the blank boxes
b) the product of the two terms that will go in the blank boxes
c) try to determine the factors of x2 + 8x +12
17. To factor x2 + 7x +12 , we know that the x2 and 12 term can be placed in the area box.
Explain what must be true about
a) the sum of the two terms that will go in the blank boxes
b) the product of the two terms that will go in the blank boxes
c) try to determine the factors of x2 + 7x +12
18. To factor x2 - 4x -12 , we know that the x2 and -12 term can be placed in the area box.
Explain what must be true about
a) the sum of the two terms that will go in the blank boxes
b) the product of the two terms that will go in the blank boxes
c) try to determine the factors of x2 - 4x - 12
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19. A number puzzle game requires players to fill in each row with the missing numbers.
Column1 is the sum from number1 and number2. Column2 is the product of the two numbers.
If you want, you can play this puzzle game with a partner alternating turns.
SUM
PRODUCT
NUMBER1
35
7
-35
7
6
8
3
-40
-5
-36
5
4
-9
18
11
24
7
-8
-10
-24
0
-36
NUMBER2
20. The first step to factor each trinomial expression is to list what the product and sum must
be. Then think of the two numbers that add and multiply to those numbers. Lastly, write out the
answer in factored form.
The first one is done for you.
Quadratic
Sum
Product
Number1
Number2
Answer
x2 + 9x + 18
9
18
3
6
(x+3 )( x+6)
x2 + 10x + 16
x2 + 2x - 15
x2 - 3x - 28
x2 - 7x + 10
x2 + 6x - 7
x2 - 2x - 24
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21. Factor each quadratic
a) x2 + 17x + 72
b) x2 - 18x + 80
c) x2 + x - 12
d) x2 - 8x + 16
e) x2 + 4x - 32
f) x2 - 11x + 10
g) x2 + 3x + 2
h) x2 + 8x + 16
i)
x2 + x - 6
k) x2 + 14x + 24
j) x2 - 5x - 24
l) x2 - 12x + 32
Special Type of factors: Difference of two perfect squares
22. Multiply together ( x + 4 )( x - 4) .
What is different about this quadratic expression and what do you think caused this ?
We can factor the expression x2 - 16 by writing it as x2 + 0x - 16. The factor numbers will have
a product of -16 and a sum of 0. The two numbers would be 4 and -4 ( x + 4) ( x - 4)
23. Factor x2 - 25
24. Factor x2 - 100
25. Why does x2 - 24 not factor ?
26. Why does x2 + 36 not factor ?
27. Make up your own difference of two perfect square binomial that will factor and then factor it
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Trinomials with the quadratic term ax2
28. Multiply the binomials together by using any method. As you multiply them, think about
where the terms are coming from.
a)
( 2x + 1) ( x + 3)
c)
(3x - 2) ( x - 4)
b) (2x - 3) ( 2x + 5)
29. Two binomials were multiplied like question 28 and the result was entered in each
rectangle. Try to determine the terms that would appear on the outside of each rectangle.
Check your answer by multiplying the terms together.
b)
a)
30. While working on problem 29, Casey noticed a pattern with the diagonals of each generic
rectangle. However, just before she shared her pattern with the rest of her team, she was called
out of class! The drawing on her paper looked like the diagram below.
a) Can you figure out what the two diagonals have in common?
Explain:
b) Does Casey’s pattern always work? Verify that her pattern works for both rectangles in
problem 29.
**IMPORTANT PLEASE READ**
The goal is to factor quadratics that are in the form 3x2 + 14x + 8. This will be a little tricky
because the product is not just 8, but 3x2 times 8 which is 24x2 . To factor 3x2 + 14x + 8 , you
will need to think of two numbers that have a sum of 14 and a product of 24. That is how the
12x and 2x will be determined in the box
Sum of 14x
and
which would be 12x and 2x
You would then continue with the process as question 29 to get the answer (3x + 2) ( x + 4)
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31. The first step to factor a quadratic equation in the form ax2 + bx + c is to think of two terms
that add together to be bx and multiply to be (ax2)(c)
For each problem, provide the sum and product that would be used in the first step of the
factoring process and then think of the two terms that have that sum and product.
Quadratic ax2 + bx + c
2x2 - 1x - 15
Sum
-1x
bx
Product (ax2)(c)
-30x2
Term1
-6x
Term2
5x
3x2 + 10x + 8
10x2 - 13x - 3
2x2 - 7x + 6
6x2 + 7x + 2
4x2 - 9x - 9
3x2 + 11x - 20
2x2 + 13x + 21
6x2 - x - 12
32.
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32. Completely factor each
a) 3x2 + 10x + 8
b) 10x2 - 13x - 3
c) 2x2 - 7x + 6
d) 6x2 + 7x + 2
e) 4x2 - 9x - 9
f) 3x2 + 11x - 20
g) 5x2 + 9x - 2
h) 2x2 + 3x + 1
i) 3x2 - 16x + 5
j) 2x2 - 11x + 5
k) 2x2 + 5x - 12
L) 4x2 - 25
Greatest Common Factor:
Sometimes there can be a greatest common factor that can be divided out of each term first to
make the coefficients smaller.
Example: 3x2 + 18x + 24
33. What number divides evenly into each term?
34. Factor that term out first from each term. Factor the remaining trinomial
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Mixed problems:
35. Factor each quadratic completely.
a) x2 + 2x - 15
b)
c) 4x2 + 4x - 3
d)
x2 - 81
e) x2 - 10x - 24
f)
3x2 - 12
g) 5x2 + 8x - 4
h) x2 - 2x - 48
i) 6x2 + 21x + 9
j) 4x2 + 5x - 6
2x2 + 2x - 12
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PART2 Solving Quadratic equations
You have studied linear equations a lot in algebra and geometry. In this section, you will be
studying parabolas. You will learn about their shape, study different equations used to graph
them and see how they can be used in real-life situations.
Congratulations! Your work at the line factory was so successful that the small local company
grew into a national corporation call Functions of America. recently your company has had
some growing pains, and new boss has turned to your team for help.
See her memo below:
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36. Your Task: Your team will be assigned its own parabola to study. Investigate your team’s
parabola and be ready to describe everything you can about it by using its graph, rule and table.
Answer the questions below to get your investigation started. You may answer them in any
order, however do not limit yourselves to these questions
• Does your parabola have any symmetry? That is can fold the graph of your parabola so that
each side of the fold exactly matches the other? If so, where would the fold line be?
• Is there a highest or lowest point on the graph of your parabola? If so, where is it? This point
is called a vertex.
• Are there any special points on your parabola? Which points do you think are important to
know?
• How would you describe the shape of your parabola? For example would you describe it as
pointing up or down?
List of Parabolas:
a) y = x 2 − 2x − 8
b) y = x 2 − 4x + 5
c) y = x 2 − 6x + 5
d) y = −x 2 + 2x − 1
e) y = −x 2 + 4
f ) y = −x 2 + 3x + 4
g) y = x 2 + 5x + 1
37. Prepare a poster for the CEO detailing your findings from your parabola
investigation. Include any insights you and your teammates found.
Explain your conclusions and justify your statements. Remember to
include a complete graph of your parabola with all special points
carefully labeled.
38. For each rule represented
to the right, state the x and y
intercepts:
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39. The graph to the right is the function y = x2 -2x - 3.
Use the graph to solve each equation
a) x2 -2x - 3 = 0
b) x2 -2x - 3 = 5
c) x2 -2x - 3 = -4
Special property of zero
40. What is 5 i 0 =
41. What is 0 i 5 =
42. What is a i 0 =
43. What is 0 i b =
44. What is (x − 3) i 0 =
45. What is 0 i (x + 1) =
IMPORTANT PLEASE READ
The special property of multiplying by zero will allow us to solve quadratic equations that
can be factored.
We are going to solve x2 -2x - 3 = 0 again, but algebraically. First determine the
factors
( x - 3) (x + 1) = 0 The number for x that makes each factor zero
will be a solution to the equation.
x-3=0
x=3
x+1=0
x = -1
Both of these numbers are solutions.
Check: place 3 in equation
(3 - 3)(3 + 1) = 0
place -1 in for equation (-1 - 3)(-1+1) = 0
0 i 4 = 0 yes
−4 i 0 = 0 yes
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We are going to solve 39b again, but algebraically
46. What was your answer for question 39 c). Notice there was only one
answer for x2 - 2x - 3 = -4?
47. You are now going to solve this equation again but algebraically. Start by setting
the equation equal to zero by adding a 4 to both sides
x2 - 2x - 3 = -4
48. Now factor this equation.
49. What number makes each factor zero? This will be the answer to the equation.
50. Using the zero product property to help you solve the quadratic equation
2x2 + 5x - 12 = 0
a. Examine the quadratic equation. We need to write this equation as a product of two
expressions. Try to factor the quadratic as previous in the unit.
b. Now that the equation is written as a product of factors equaling zero, you can use the zero
product property on each factor to solve it. Since you know that one of the factors must be
zero, you can set up two smaller equations to help you sole for x. Use one factor at a time
and determine what x-value makes it equal to zero.
c. You now know the roots of the equation 2x2 + 5x - 12 = 0 ( also called the zero). Use a
graphing app or calculator to graph 2x2 + 5x - 12 to verify these special points.
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51. Solve each equation algebraically.
a. (x-2)(x+8) = 0
d. (x - 7) 2 = 0
b. (3x - 9) (x - 1) = 0
c. (x + 10)(2x - 5) = 0
e. x2 + x - 6 = 0
x2 + 2x = 24
g. x2 = 3x + 10
h. 2x2 + x - 15 = 0
i. 3x2 - 10x = 8
f.
j. 2x2 + 6x + 4 = 0
k. x2 - 49 = 0
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52. The graph for
y = x2 + x - 6 is given to the right.
a) In question 51e you solved the equation x2 + x - 6 = 0.
Mark your x-value solutions from question 51e on the graph.
Notice where they are located.
b) So when you solve the equation x2 + x - 6 = 0
you are also finding what key feature of the graph?
53. The graph for y = ( x - 2 )(x + 4) is to the right.
Locate the x-intercepts. How could you determine
the x-intercepts from the equation without looking
at the graph?
54. What would be the x-intercepts for the graph y = (2x + 5)( x- 6) ?
55. Find the x-intercepts for each quadratic equation below without looking at the graph
a. y = x2 + 6x + 8
b. y = 3x2 - 7x + 4
c. y = (x+5)(-2x + 3)
d. y = x2 + 6x
e. y = 3(x - 5)(2x + 3)
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56. a) Try to factor 2x2 - 3x - 4
b) If it is not factorable, does that mean that 2x2 - 3x - 4 =0
does not have any solutions?
c) Make a sketch of y = 2x2 - 3x - 4 on the graph
to the right.
d) estimate the x-intercepts which are also the
solutions to 2x2 - 3x - 4 = 0
Quadratic formula:
−b ± b 2 − 4ac
x=
2a
57. Since a parabola can have x-intercepts even when its corresponding quadratic equation is
not factorable, another way to find the roots of a quadratic equation is needed.
a. One way to find the roots of a quadratic equation is by using the Quadratic Formula, shown
below. This formula uses values a,b and c from a quadratic equation written in standard
form. When the quadratic equation is written in standard form ( looks like: 2x2 - 3x - 4 = 0)
then a is the number of x2 terms, b is the number of x terms and c is the constant.
If 2x2 - 3x - 4 = 0 then what are
a ______ b _______ c _______?
b. The quadratic Formula calculates two possible answers by using the ± symbol. This
symbol (read as “plus or minus”) is shorthand notation that tells you to calculate the formula
twice: once with addition and once with subtraction in the numerator. Quadratic Formula
problem is really two different problems unless the value of b 2 − 4ac = 0.
Carefully substitute a,b,and c from 2x2 - 3x - 7 = 0 into the Quadratic Formula. Evaluate each
expression (once with addition and once with subtraction) to solve for x.
CLASS NOTES QUADRATIC FORMULA
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58. Watch the videos (These can be found on the class webpage)
a. http://www.virtualnerd.com/tutorials/?id=Alg1_12_01_0007
b. https://learnzillion.com/lessons/748-solve-a-quadratic-using-the-quadratic-formula
c. http://www.phschool.com/atschool/academy123/english/academy123_content/wl-bookdemo/ph-298s.html
59. Use the quadratic Formula to solve each quadratic equation
a) x2 + 5x + 1 =0
b) x2 - 4x + 2 = 0
c) 2x2 + 2x - 4 = 0
d) 3x2 - 2x - 5 = 0
−b ± b 2 − 4ac
x=
2a
e) -x2 + x + 6 = 0
60. Graph each of the quadratic equations in question #59 to check your answers.
61. For the quadratic equation 6x2 + 11x - 10 = 0
a. Solve it using the zero product property.
b. Solve it using the Quadratic Formula.
c. Did the solution from parts (a) and (b) match? If not, why not?
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62. The quadratic Formula can solve any quadratic equation ax2 + bx + c = 0 if a 0. But what
if the equation is not in standard form? What if terms are missing? Consider these questions
as you solve the quadratic equations below. Share your ideas with your teammates and be
prepared to demonstrate your process for the class.
a)
c)
4x2 - 121 = 0
15x2 - 165x = 630
b) 2x2 - 2 - 3x = 0
d) 36x2 + 25 = 60x
63. The Saint Louis Gateway Arch
The Saint Louis Gateway Arch has a shape much like a parabola. Suppose the Gateway Arch
can be approximated by y = 630 - 0.00635x2 , where both x and y represent distances in feet
and the origin is the point on the ground directly below the arch’s apex(highest point)
a) Find the x-intercept of the Gateway Arch.
What does this information tell you?
b) How wide is the arch at its base?
c) How tall is the arch? How did you find your solution?
d) draw a quick sketch of the arch labeling the
axes with all the the values you know.
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THE DISCRIMINANT
The discriminant of the quadratic formula is the part
2x2 - 3x - 6 , the discriminant would be
b 2 − 4ac . If the quadratic was
(−3)2 − 4(2)(−6) = 57
64. Solve each equation by using the quadratic formula, state the discriminate (i.e. 20 ) and
then make a rough sketch of the parabola.
a)
2x2–5x+2=0b.x2+4x+4=0
a=_______b=________c=_______a=_______b=________c=_______
Discriminant=_________Discriminant=_________
c)
-x2+3x+2=0d.2x2-3x+4=0
a=_______b=________c=_______a=_______b=________c=_______
Discriminant=_________Discriminant=_______
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e)
4x2-12x+9=0f.x2+2x+5=0
a=_______b=________c=_______a=_______b=________c=_______
Discriminant=_________Discriminant=_________
65.Whichequationsdidnothaveanysolution?Howcanyoutellfromthegraphtherewill
benosolutions?Howcanyoutellfromthequadraticformulatherewillbenosolutions.
66.Whichequationsonlyhadonesolution?Howcanyoutellfromthegraphitwillonly
haveonesolution?Howcanyoutellfromthequadraticformulatherewillbeonlyone
solution?
67.Whichequationshadtwosolutions?Howcanyoutellfromthegraphitwillhavetwo
solutions?Howcanyoutellfromthequadraticformulatherewillbetwosolutions?
Choosing a Strategy: Which method should I use?
You now have two algebraic methods to solve quadratic equations: using the zero product
property and using the quadratic formula. How can you decide which strategy is best to try
first? You should have some strategies to help you determine which method to try first when
solving a quadratic equation.
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68. Examine the quadratic equations below with your team. For each equation:
• decide which strategy is best to try first.
• Solve the equation. If your first strategy does not work, switch to the other strategy.
• Check your solutions.
a) x2 + 12x + 27 = 0
b) 0.5x2 + 9x + 3.2 = 0
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c) (3x+4)(2x - 1) = 0
d) x2 + 16x = 8x
e) x2 + 5 - 2x = 0
f. 20x2 - 30x = 2x + 45
69. While solving (x - 5) (x + 2) = -6 , Kyle decided that x must equal 5 or -2. “Not so fast!”
exclaimed Stanton. “The product does not equal zero. We need to change the equation first.”
a) What is Stanton talking about?
b) How can the equation be rewritten? Write the equation in standard form.
c) Solve the resulting equation for x. Does your solutions match Kyle’s answer of 5 or -2 ?
70. Joe is playing with a yo-yo. He throws the yo-yo down and then pulls it back up. The
motion of the yo-yo is represented by the equation y = 2x2 - 4.8x, where x represents the
number of seconds since the yo-yo left Joe’s hand and y represents the vertical height in inches
of the yo-yo with respect to Joe’s hand. Note that when the yo-yo is in Joe’s hand, y=0 and
when the yo-yo is below his hand, y is negative
a) How long is Joe’s yo-yo in the air before it comes back to Joe’s hand?
Write and solve a quadratic equation to find the times that the yo-yo
is in Joe’s hand.
b) At what time does the yo-yo turn around? Use what you know about
parabolas to help you.
c) How long is the yo-yo string? That is what is y when the yo-yo changes
direction?
d) Draw a sketch of the graph representing the motion of Joes’ yo-yo.
Label important points
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Use any method to solve each quadratic equation. Some of the quadratics might not have a
solution. If it doesn’t indicate “no Solution”
71. (4x - 7) (2x + 3) = 0
72. x2 = 16
73. 4x2 - 36 = 0
74. x2 + 4x - 8 = 0
75. -2x2 + 5x = -12
76. 2x2 + 3x + 5 = 0
77. x2 = 4x - 1
78. x2 + 8x + 16 = 0
79. x2 - 4x = -10
80. x2 - 10x = -25
81. The quadratic equation in the form ax2 + bx + c = 0 was graphed below. In each case, what
are the solutions(x-intercepts) to the quadratic?
b)
a)
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