Download Secondary II - Northern Utah Curriculum Consortium

Secondary II
Solving Quadratic Equations
Teacher Edition
Unit 5
Northern Utah Curriculum Consortium
Project Leader
Sheri Heiter
Weber School District
Project Contributors
Ashley Martin
Bonita Richins
Craig Ashton
Davis School District
Cache School District
Cache School District
Gerald Jackman
Jeff Rawlins
Jeremy Young
Box Elder School District
Box Elder School District
Box Elder School District
Kip Motta
Marie Fitzgerald
Mike Hansen
Rich School District
Cache School District
Cache School District
Robert Hoggan
Sheena Knight
Teresa Billings
Cache School District
Weber School District
Weber School District
Wendy Barney
Helen Heiner
Susan Summerkorn
Weber School District
Davis School District
Davis School District
Lead Editor
Allen Jacobson
Davis School District
Technical Writer/Editor
Dianne Cummins
Davis School District
NUCC| Secondary II Math i
Table of Contents
5.1 UNDERSTANDING SQUARES ..................................................................................................................1
Teacher Notes ..................................................................................................................................................1
Mathematics Content .......................................................................................................................................4
Understanding Squares A Develop Understanding Task 1 ..............................................................................5
Ready, Set, Go! ................................................................................................................................................6
Solutions: .........................................................................................................................................................8
5.2 SOLVE USING SQUARE ROOTS ..............................................................................................................9
Teacher Notes ..................................................................................................................................................9
Mathematics Content .....................................................................................................................................12
Gravity A Solidify Understanding Task 2 ......................................................................................................13
Ready, Set, Go! ..............................................................................................................................................14
Solutions: .......................................................................................................................................................16
5.3 SOLVE BY FACTORING WHEN a=1 ......................................................................................................17
Teacher Notes ................................................................................................................................................17
Mathematics Content .....................................................................................................................................21
Algebra Tiles A Develop Understanding Task 3 ...........................................................................................22
Ready, Set, Go! ..............................................................................................................................................23
Solutions: .......................................................................................................................................................25
5.4 QUADRATIC EQUATIONS BY FACTORING WHEN a >1 ...................................................................26
Teacher Notes ................................................................................................................................................26
Mathematics Content .....................................................................................................................................29
Area of a Painting A Practice Understanding Task 4 ....................................................................................30
Ready, Set, Go! ..............................................................................................................................................31
Solutions: .......................................................................................................................................................33
5.5 SOLVE BY COMPLETING THE SQUARE ..............................................................................................34
Teacher Notes ................................................................................................................................................34
Mathematics Content .....................................................................................................................................38
Garden Space A Develop Understanding Task 5 ...........................................................................................39
Ready, Set, Go! ..............................................................................................................................................40
Solutions: .......................................................................................................................................................43
NUCC| Secondary II Math ii
5.6 SOLVE WITH COMPLEX NUMBERS .....................................................................................................44
Teacher Notes ................................................................................................................................................44
Mathematics Content .....................................................................................................................................48
Review A Solidify Understanding Task 6 ......................................................................................................49
Ready, Set, Go! ..............................................................................................................................................50
Solutions: .......................................................................................................................................................53
5.7 SOLVE USING THE QUADRATIC FORMULA ......................................................................................54
Teacher Notes ................................................................................................................................................54
Mathematics Content .....................................................................................................................................58
Analyze Graphs of Quadratic Functions A Develop Understanding Task 7 ..................................................59
Ready, Set, Go! ..............................................................................................................................................60
Solutions: .......................................................................................................................................................63
H5.8 FUNDAMENTAL THEOREM OF ALGEBRA ......................................................................................64
Teacher Notes ................................................................................................................................................64
Mathematics Content .....................................................................................................................................69
Investigating the Discriminant A Develop Understanding Task 8 .................................................................70
Ready, Set, Go! ..............................................................................................................................................72
Solutions: .......................................................................................................................................................75
5.9 OPTIONAL CATAPULT LESSON............................................................................................................76
Teacher Notes ................................................................................................................................................76
Mathematics Content .....................................................................................................................................79
Shape of Quadratic Function A Practice Understanding Task 9 ...................................................................80
Ready, Set, Go! ..............................................................................................................................................81
Solutions: .......................................................................................................................................................82
NUCC| Secondary II Math iii
Unit 5.1
5.1 UNDERSTANDING SQUARES
Teacher Notes
Time Frame: 1 class period (90 minutes)
Materials Needed: Rulers
Purpose: Students will understand the concept of a square root. They will also learn to simplify
radical expressions including rationalizing the denominator.
Core Standards Focus: A.REI.4 Solve quadratic equations by inspection, taking square roots,
completing the square, the quadratic formula, and factoring, as appropriate to the initial form of
the equation. Recognize when the quadratic formula gives complex solutions and write them as
a + bi for real numbers a and b.
Related Standards: Perimeter and area of a square. Solving equations.
Launch (Whole Class): Hand out the rulers and the task worksheet. Allow students 10 minutes
to complete. Discuss their answers.
Draw a square on the board and write 64 on the inside of the square. If 64 is the area, what is the
length of one side? How did you find the answer? Change the 64 to 100. What is the side length?
Why didn’t you just divide 100 by 4?
NUCC | Secondary II Math 1
Unit 5.1
Explore (Individual, small group or pairs):
Divide students into groups of 4. Have 1 student in the group be the ‘scribe’ and write down the
group’s answers.
On the board change the square’s area to 18. Now what is the side length? How could you write
this in radical notation? Draw a square with the area of 18 units. Draw in the units.
If time, change the area to 200 and have students discuss the side length and how to draw the
units.
Discuss (Whole Class or Group):
Demonstrate simplifying a radical expression.
√18= √9  2 = 3√ 2
√200= √100  2 = 10√2
Now check to see if √18 = 3√2
4.24 = 4.24
A number ‘r’ is a square root of a number ‘s’ if r2 = s.
The expression √𝑠 is called a radical. The symbol√ is a radical sign and the number s
beneath the radical sign is the radicand of the expression.
Example 1:
a.
Simplify the expressions. (Use properties of square roots)
√8 = √4  2 = √4  √2 = 2 √2
b. √8  √6 = √48 = √16  √3 = 4 √3
Remind students to refer to the root/square chart they created in the task for assistance.
5
c. √36 =
√5
√36
=
√5
6
Example 2: Rationalize denominators of fractions
7
𝑎. √3 =
√7
√3
=
√7
√3

√3
√3
=
√21
3
b. Introduce ‘conjugate’. The conjugate of 5 + √2 is 5 - √2 . When multiplying conjugates
together, the answer is a real, rational number.
(5 + √2 )( 5 - √2) = 25 - 5√2 + 5√2 - √2 √2 = 25 – 2 = 23
NUCC | Secondary II Math 2
Unit 5.1
𝑐.
5
3−√7
5
3+√7
= 3−√7  3+√7 =
5(3+√7)
9−7
=
15+5√7
2
Assignment: Ready, Set, Go!
NUCC | Secondary II Math 3
Unit 5.1
Mathematics Content
Cluster Title: Solve equations and inequalities in one variable.
Standard A.REI.4: Solve quadratic equations in one variable.
a. Use the method of completing the square to transform any quadratic equation in x into an
equation of the form (x-p) 2 = q that has the same solutions. Derive the quadratic formula
from this form
b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing
the square, the quadratic formula and factoring, as appropriate to the initial form of the
equation. Recognize when the quadratic formula gives complex solutions and write them as a
 bi for real numbers a and b.
Concepts and Skills to Master
 Complete the square.
 Solve quadratic equations, including complex solutions, using completing the square,
quadratic formula, factoring, and by taking the square root.
 Derive the quadratic formula from completing the square.
 Recognize when one method is more efficient than the other.
 Interpret the discriminant.
 Understand the zero product property and use it to solve a factorable quadratic equation.
Critical Background Knowledge
 Factor
 Simplify radicals
 Understanding of complex numbers (Secondary II: N.CN.1)
 Understand the real number and complex number systems (Secondary II: N.CN.1)
Academic Vocabulary
radicals, complex numbers solve, factor, discriminant
Suggested Instructional Strategies
 Use of algebra tiles to demonstrate simple completing the square problems (see NCTM
MTMS, March 2007, p. 403).
Skills Based Task:
Solve the equation 6x2 –x –15 = 0 by
factoring and by completing the square.
Justify each method using mathematical
properties.
Problem Task:
Solve the quadratic equation 49x2 – 70x +37 = 0
using two methods. Describe the advantages of
each method.
Some Useful Websites:
Illuminations: Proof Without Words: Completing the Square
http://illuminations.nctm.org/ActivityDetail.aspx?ID=132
NUCC | Secondary II Math 4
Unit 5.1
Understanding Squares
A Develop Understanding Task 1
Name_____________________________________
Hour___________
1. Use a ruler and draw a square in the space below.
2. How did you draw the square?
3. Why is this shape a square?
4. What is the area of the square above?
5. Refer to the square that was drawn above and ‘show’ the area of the square.
6. Complete the following chart of roots, squares, and square roots for the numbers 1-20.
Root
Square
Square Root
1
1
1
2
4
2
NUCC | Secondary II Math 5
Unit 5.1
Ready, Set, Go!
Name__________________________________________________
Hour____________
Ready
Simplify the expression.
1. √ 200
2. √ 147
3. √ 5  √ 50
4. √121
5.
13
2
5+√3
Set
Simplify the expression.
1. √18
2. √ 48
3. √20
4. √ 98
5. √ 12  √ 7
6. √ 9  3√ 27
25
7. √16
49
8. √ 9
Go!
Simplify the expression.
100
2. √ 1
49
4. √5
45
6. √5  √5
1. √ 25
3. √ 9
5. √ 3
27
7
4
3
NUCC | Secondary II Math 6
Unit 5.1
7. What is the conjugate of 7 +√2 ? What is the answer when these conjugates are multiplied
together?
8. Why is it necessary to multiply the denominator by its conjugate?
9.
1
10.
3 + √3
1+ √5
11. 5 +
2
5 − √3
2 − √2
12. 3 +
√5
√7
13. List all possible digits that occur in the units’ place of the square of a positive integer.
(Hint: Look at the root/square chart that was created earlier.) Use this list of digits to
determine whether or not it is possible for each root to be an integer.
Guess
Calculator Answer
Is this a perfect square?
14. √4527
15. √5233
16. √11,286
17. √272,484
18. √1,500,378
19. Is this a good rule for square roots? Explain.
20. Give an example of a 5-digit perfect square and its root.
NUCC | Secondary II Math 7
Unit 5.1
Solutions:
NUCC | Secondary II Math 8
Unit 5.2
5.2 SOLVE USING SQUARE ROOTS
Teacher Notes
Time Frame: 1 class period (90 minutes)
Materials Needed: Pencils
Purpose: Students will learn how to solve quadratic equations using square roots while
using ± notation and to simplify the radicals when necessary.
Core Standards Focus: A.REI.4 Solve quadratic equations by inspection, taking square roots,
completing the square, the quadratic formula, and factoring, as appropriate to the initial form of
the equation. Recognize when the quadratic formula gives complex solutions and write them as
a + bi for real numbers a and b.
Related Standards: Simplifying radicals, solving linear equations
Launch (Whole Class): Hand out the Task worksheet. Allow students 10 minutes to complete.
Explore (Pair/Share):
Assign each student a partner to share their answers and to compare the way they solved question
# 9. Have students show various ways that they solved the equation.
NUCC | Secondary II Math 9
Unit 5.2
Also, explain that the equation of gravity is: h = - 16t2 + h0
What does h represent? (ending height, usually 0) t? (time, usually seconds) h0? (initial height)
Solve the equation when the initial height is 324.
Discuss (Whole Class or Group):
Ask: ‘How could you solve this equation x2 = 64?’
Ask: What numbers could you put in for x that would make that statement true?
Show - 8 would also work.
To solve quadratic equations, first isolate the x2 term. Then take the square root of both sides of
the equation. Show ± notation for the positive and negative solutions.
Example 1:
a. x2 = 100
x = ± 10
b. 2x2 – 3 = 87
2x2 = 90
x2 = 45
x = ±√45
x = ±√9  √5
x = ± 3√5
Add 3 to each side
Divide both sides by 2
Take square roots of each side.
Product property
*Simplify.
c. Finding solutions of a quadratic equation
Find the solutions of ½ (w - 2)2 + 1 = 4
½ (w – 2)2 + 1 = 4
(w – 2)2 = 6
Subtract 1 from both sides and multiply both sides
by 2.
w – 2 = ±√6
Take square roots of each side.
w = 2 ± √6
Add 2 to each side
The solutions are 2 + √6 and 2 - √6
*Remember to simplify the radicals and rationalize the denominator when necessary.
NUCC | Secondary II Math 10
Unit 5.2
Assignment: Ready, Set, Go!
NUCC | Secondary II Math 11
Unit 5.2
Mathematics Content
Cluster Title: Solve equations and inequalities in one variable.
Standard A.REI.4: Solve quadratic equations in one variable.
a. Use the method of completing the square to transform any quadratic equation in x into an
equation of the form (x-p) 2 = q that has the same solutions. Derive the quadratic formula
from this form
b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing
the square, the quadratic formula and factoring, as appropriate to the initial form of the
equation. Recognize when the quadratic formula gives complex solutions and write them as a
 bi for real numbers a and b.
Concepts and Skills to Master
 Complete the square.
 Solve quadratic equations, including complex solutions, using completing the square,
quadratic formula, factoring, and by taking the square root.
 Derive the quadratic formula from completing the square.
 Recognize when one method is more efficient than the other.
 Interpret the discriminant.
 Understand the zero product property and use it to solve a factorable quadratic equation.
Critical Background Knowledge
 Factor
 Simplify radicals
 Understanding of complex numbers (Secondary II: N.CN.1)
 Understand the real number and complex number systems (Secondary II: N.CN.1)
Academic Vocabulary
radicals, complex numbers solve, factor, discriminant
Suggested Instructional Strategies
 Use of algebra tiles to demonstrate simple completing the square problems (see NCTM
MTMS, March 2007, p. 403).
Skills Based Task:
Solve the equation 6x2 –x –15 = 0 by
factoring and by completing the square.
Justify each method using mathematical
properties.
Problem Task:
Solve the quadratic equation 49x2 – 70x +37 = 0
using two methods. Describe the advantages of
each method.
Some Useful Websites:
Illuminations: Proof Without Words: Completing the Square
http://illuminations.nctm.org/ActivityDetail.aspx?ID=132
NUCC | Secondary II Math 12
Unit 5.2
Gravity
A Solidify Understanding Task 2
Name_____________________________________
Hour___________
Simplify the expressions.
1. √196
2. √250
3. √16  √4
4.
√25
√5
5. Give an example of a binomial and its conjugate.
6. What is the conjugate of 2 - √5 ?
7. Simplify
3
2− √5
8. Write down what you know about the Law of Gravity.
9. The height h (in feet) of a ball dropped from the top of a building can be
modeled by h = - 16t2 + 256 where t is the time (in seconds). Solve the equation
to find the time it takes for the ball to hit the ground, or when h = 0.
NUCC | Secondary II Math 13
Unit 5.2
Ready, Set, Go!
Name__________________________________________________
Hour____________
Ready
Solve the equation.
1. x2 = 16
2. x2 = 144
3. x2 = 121
4. x2 – 4 = 0
5. x2 – 64 = 0
Set
6. x2 – 8 = 0
7. 2x2 = 32
8. 3x2 = 75
9.
10. x2 + 12 = 13
11. x2 – 1 = 6
1 2
x
3
= 12
12. 20 – x2 = - 29
Go!
13. 2x2 - 1 = 7
14. 3x2 – 9 = 0
15. ½ x2 + 8 = 17
NUCC | Secondary II Math 14
Unit 5.2
When an object is dropped, its height h can be determined after t seconds by using the falling
object model h = -16 t2 + s where s is the initial height. Find the time it takes an object to hit the
ground when it is dropped from a height of s feet.
16. s = 100
17. s = 196
18. s = 480
19. s = 600
20.
21. s = 1200
s = 750
22. From 1970 – 1990, the average cost of a new car C (in dollars) can be approximated by the
model C = 30.5t2 + 4192 where t is the number of years since 1970. During which year was
the average cost of a new car $7242?
Write a quadratic equation that has the given solutions.
23. ± √5
24. ± 3√2
25. -1 ± √6
NUCC | Secondary II Math 15
Unit 5.2
Solutions:
NUCC | Secondary II Math 16
Unit 5.3
5.3 SOLVE BY FACTORING WHEN a=1
Teacher Notes
Time Frame: 1 class period (90 minutes)
Materials Needed: Algebra Tiles – actual tiles, paper cut out tiles, or demonstrate with your
computer using the site: http://illuminations.nctm.org/ActivityDetail.aspx?ID=216 or any other
site you may find.
Purpose: Solve quadratic equations by factoring where a = 1. Students have been introduced to
factoring in Secondary Math 1, but they may need some review.
Core Standards Focus: A.REI.4 Solve quadratic equations by inspection, taking square roots,
completing the square, the quadratic formula, and factoring, as appropriate to the initial form of
the equation. Recognize when the quadratic formula gives complex solutions and write them as a
+ bi for real numbers a and b.
Related Standards: Factoring quadratics where a = 1
Launch (Whole Class): Distribute the Algebra Tiles Task worksheet to each student. Have them
work on this for 10 minutes.
Remember factoring? What does it mean to factor a quadratic equation?
NUCC | Secondary II Math 17
Unit 5.3
Explore (Individual, small group or pairs):
Have students share their task worksheet with another student. They can explain what their
Algebra Tile diagram looks like. Have them draw one more equation on their paper and factor it.
x2 + 3x + 2
Discuss (Whole Class or Group):
Set each of the quadratic equations from the task worksheet equal to 0. (Have the equations in
quadratic form first.)
How do you think we could solve for x? (Let them discover that factoring would be a good way.)
Demonstrate that when the equation = 0, then set each factor = 0 and solve each one separately.
There may be 0, 1, or 2 solutions.
Have students factor the following and help them recognize the pattern with ‘c’ in each
expression. (Remind students of the standard form y = ax2 + bx + c).
Example 1:
x2 + 7x + 12 =
x2 – 7x + 12 =
(x + 3)(x + 4) notice c > 0
(x – 3)(x – 4)
x2 + x – 12 =
x2 – x - 12 =
(x – 3)( x + 4) notice c < 0
(x + 3)( x – 4)
a. Factor trinomials
w2 – 2w – 15
= (w + 3)(w – 5)
b. Special Patterns
g2 – 20g + 100
(Perfect square trinomial)
= (g – 10)(g – 10)
= (g – 10)2
z2 – 64
(Difference of 2 squares)
2
2
=z -8
= (z + 8)(z – 8)
After each equation is factored, then set each factored term equal to 0. Solve each factored term
and those answers will be the solutions (also called zeros and roots).
x2 + 7x + 12 =
(x + 3)(x + 4) = 0
so, x + 3 = 0 and x + 4 = 0
x = -3
and x = -4
NUCC | Secondary II Math 18
Unit 5.3
Example 2:
Find the roots of an equation
Find the roots of x2 – 13x + 42 = 0
x2 – 13x + 42 = 0
(x – 6)(x – 7) = 0
x – 6 = 0 or x – 7 = 0
x=6
or x = 7
Example 3:
Factor
Set each factor = 0
Solve for x
Use a quadratic equation as a model.
A rectangular garden measures 10 feet by 15 feet. By adding x feet to the width and x
feet to the length, the area is doubled. Find the new dimensions of the garden.
New area
2 (10)(15)
300
0
0
x + 30 = 0
x = -30
=
=
=
=
=
or
or
New length
(10 + x)
150 + 25x + x2
x2 + 25x – 150
(x + 30)(x – 5)
x–5=0
x=5


New width
(15 + x)
Multiply using FOIL
Write in standard form
Factor
Set each factor = 0
Solve for x
Reject the negative value. The garden’s width and length should each be increased by 5
feet. The new dimensions are 15 feet by 20 feet.
Example 4:
Find the zeros (solutions) of quadratic functions
Find the zeros of y = x2 + 3x – 28
y = x2 + 3x – 28
= (x + 7)(x – 4)
x + 7 = 0 or x – 4 = 0
x = -7 or x = 4
Factor
Set each factor = 0
Solve for x
NUCC | Secondary II Math 19
Unit 5.3
Assignment: Ready, Set, Go!
NUCC | Secondary II Math 20
Unit 5.3
Mathematics Content
Cluster Title: Solve equations and inequalities in one variable.
Standard A.REI.4: Solve quadratic equations in one variable.
a. Use the method of completing the square to transform any quadratic equation in x into an
equation of the form (x-p) 2 = q that has the same solutions. Derive the quadratic formula
from this form
b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing
the square, the quadratic formula and factoring, as appropriate to the initial form of the
equation. Recognize when the quadratic formula gives complex solutions and write them as a
 bi for real numbers a and b.
Concepts and Skills to Master
 Complete the square.
 Solve quadratic equations, including complex solutions, using completing the square,
quadratic formula, factoring, and by taking the square root.
 Derive the quadratic formula from completing the square.
 Recognize when one method is more efficient than the other.
 Interpret the discriminant.
 Understand the zero product property and use it to solve a factorable quadratic equation.
Critical Background Knowledge
 Factor
 Simplify radicals
 Understanding of complex numbers (Secondary II: N.CN.1)
 Understand the real number and complex number systems (Secondary II: N.CN.1)
Academic Vocabulary
radicals, complex numbers solve, factor, discriminant
Suggested Instructional Strategies
 Use of algebra tiles to demonstrate simple completing the square problems (see NCTM
MTMS, March 2007, p. 403).
Skills Based Task:
Solve the equation 6x2 –x –15 = 0 by
factoring and by completing the square.
Justify each method using mathematical
properties.
Problem Task:
Solve the quadratic equation 49x2 – 70x +37 = 0
using two methods. Describe the advantages of
each method.
Some Useful Websites:
Illuminations: Proof Without Words: Completing the Square
http://illuminations.nctm.org/ActivityDetail.aspx?ID=132
NUCC | Secondary II Math 21
Unit 5.3
Algebra Tiles
A Develop Understanding Task 3
Name_____________________________________
Hour___________
1. Describe what it means to ‘factor a quadratic equation’.
2. Draw the algebra tiles: x, x2, and 1.
3. Draw what the quadratic equation x2 + 5x + 6 would look like using Algebra Tiles.
4. Write down the length and width of the above tile diagram. What does this represent?
5. Draw what the quadratic equation x2 + 8x + 16 would look like using Algebra Tiles.
6. Write down the length and width of the above tile diagram. What does this represent?
NUCC | Secondary II Math 22
Unit 5.3
Ready, Set, Go!
Name__________________________________________________
Hour____________
Ready
Factor the expression. If the expression cannot by factored, say so.
1.
x2–x–2
2. x2 – 4x + 3
3. x2 + 8x + 15
4. x2 – 4
5. x2 + 2x + 1
6. x2 – 10x + 25
Set
Solve the equation.
x2 – 2x – 3 = 0
8. x2 + 3x + 2 = 0
9. x2 – 2x + 1 = 0
10. x2 + 4x + 4 = 0
11. x2 – 9x + 14 = 0
12. x2 – 49 = 0
13. x2 – 4x = 12
14. x2 = 64
7.
NUCC | Secondary II Math 23
Unit 5.3
Go!
Find the zeros of the functions
15. y = x2 + x – 20
16. y = x2 - 9
17. f(x) = x2 – 7x + 6
18. g(x) = x2 + 7x + 10
19. y = x2 – 6x – 7
20. h(x) = x2 + 5x – 24
21. y = x2 + 3x
Find the value of x.
22. Area of the rectangle = 28
23. Area of the rectangle = 32
x
x-4
x
x+3
NUCC | Secondary II Math 24
Unit 5.3
Solutions:
Space saver for answers/solutions for unit 5.3
NUCC | Secondary II Math 25
Unit 5.4
5.4 QUADRATIC EQUATIONS BY FACTORING
WHEN a >1
Teacher Notes
Time Frame: 1 class period (90 minutes)
Materials Needed: Pencils
Purpose: Students will discover how to solve quadratic equations by factoring when a > 1.
Core Standards Focus: A.REI.4 Solve quadratic equations by inspection, taking square roots,
completing the square, the quadratic formula, and factoring, as appropriate to the initial form of
the equation. Recognize when the quadratic formula gives complex solutions and write them as a
+ bi for real numbers a and b.
Related Standards: Factoring quadratic equations when a = 1
Launch (Whole Class): Put students in groups of 3 and give each group the task worksheet.
Allow groups 5 – 10 minutes to solve.
Explore (Pair/Share): Have one student from each group come to the board and show how they
solved the task.
NUCC | Secondary II Math 26
Unit 5.4
Discuss (Whole Class or Group):
Which way (referring to the board examples) seems the most understandable to you?
Look at these quadratic equations and identify the similarities and differences.
2x2 – x – 3 = 0
x2 + 2x + 1 = 0
Now factor both equations and solve.
Which equation was easier to factor and why?
In the previous lesson, we factored quadratic equations where the ‘a’ term was always 1.
Now we will factor quadratic equations where a > 1.
There are different techniques to factoring these equations. But they all include the very first step
which is to factor out the GCF (greatest common factor) if there is one.
Teachers: Please feel free to show any methods that you wish. I prefer the ‘guess and check’
method because they learn more about factors when they look at the factors of the ‘a’ and ‘c’
terms and eventually can learn to make educated guesses.
Example 1:
Factor 2x2 – x – 3.
(Show your method.)
Answer (2x – 3)(x + 1)
Example 2: Factor with special patterns.
Factor the expression.
a. 6t2 – 24 = 6 (t2 – 4)
= 6 (t – 2) (t + 2)
b.
Factor out GCF
Difference of 2 squares
3m2 – 18m + 27 = 3 (m2 – 6m + 9)
= 3 (m – 2 )2
Factor out GCF
Perfect square trinomial
Example 3: Solve a quadratic equation.
4s2 + 11s + 8 =
4s2 + 8s + 4 =
s2 + 2s + 1 =
(s + 1) 2 =
s+1 =
s=
3s + 4
0
0
0
0
-1
Write in standard form
Divide each side by 4
Factor
Set factor = 0
Solve for s.
NUCC | Secondary II Math 27
Unit 5.4
Assignment: Ready, Set, Go!
NUCC | Secondary II Math 28
Unit 5.4
Mathematics Content
Cluster Title: Solve equations and inequalities in one variable.
Standard A.REI.4: Solve quadratic equations in one variable.
a. Use the method of completing the square to transform any quadratic equation in x into an
equation of the form (x-p) 2 = q that has the same solutions. Derive the quadratic formula
from this form
b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing
the square, the quadratic formula and factoring, as appropriate to the initial form of the
equation. Recognize when the quadratic formula gives complex solutions and write them as a
 bi for real numbers a and b.
Concepts and Skills to Master
 Complete the square.
 Solve quadratic equations, including complex solutions, using completing the square,
quadratic formula, factoring, and by taking the square root.
 Derive the quadratic formula from completing the square.
 Recognize when one method is more efficient than the other.
 Interpret the discriminant.
 Understand the zero product property and use it to solve a factorable quadratic equation.
Critical Background Knowledge
 Factor
 Simplify radicals
 Understanding of complex numbers (Secondary II: N.CN.1)
 Understand the real number and complex number systems (Secondary II: N.CN.1)
Academic Vocabulary
radicals, complex numbers solve, factor, discriminant
Suggested Instructional Strategies
 Use of algebra tiles to demonstrate simple completing the square problems (see NCTM
MTMS, March 2007, p. 403).
Skills Based Task:
Solve the equation 6x2 –x –15 = 0 by
factoring and by completing the square.
Justify each method using mathematical
properties.
Problem Task:
Solve the quadratic equation 49x2 – 70x +37 = 0
using two methods. Describe the advantages of
each method.
Some Useful Websites:
Illuminations: Proof Without Words: Completing the Square
http://illuminations.nctm.org/ActivityDetail.aspx?ID=132
NUCC | Secondary II Math 29
Unit 5.4
Area of a Painting
A Practice Understanding Task 4
Name_____________________________________
Hour___________
1. The area of a painting is 24 square inches and the length is 5 inches more than the width.
Find the length of the painting. Show all your work.
2. The area of a painting is 14 square feet and the width is 5 inches less than the length. Find the
width of the painting. Show all your work.
NUCC | Secondary II Math 30
Unit 5.4
Ready, Set, Go!
Name__________________________________________________
Hour____________
Ready
Factor the expression. If the expression cannot be factored, say so.
1. 2x2 + 5x + 2
2. 2x2 – 3x + 1
3. 5x2 + x – 4
4. 6x2 + 2x – 4
5. 2x2 – 10x + 12
6. 8x2 – 20x – 12
Set
Solve the equation.
7. 2x2 – 3x + 1 = 0
8. 2x2 + 5x + 3 = 0
9. 6x2 – 7x + 2 = 0
10. 3x2 – 8x – 3 = 0
11. 4x2 – 7x + 3 = 0
12. 4x2 – 4x – 15 = 0
13. 2x2 – 2x – 12 = 0
14. 6x2 – 15x – 9 = 0
15. 12x2 + 4x – 8 = 0
NUCC | Secondary II Math 31
Unit 5.4
Go!
Find the solutions of the function. Remember to set it = 0 first.
16. y = 2x2 – 6x + 4
17. y = 3x2 + 6x – 9
18. f(x) = 5x2 + 10x – 40
19. y = 4x2 – 12x + 9
20. g(x) = 18x2 – 2
21. y = 16x2 + 64x + 60
Find the value of x.
22. Area of the square = 81
3x
3x
23. Area of the rectangle = 16
x
3x + 2
24. Multiple Choice What are all the solutions to 2x2 + 3x + 6 = x2 + 3x + 15?
A. 3
B. -3, 3
C. -3
D. 2, 4
25. A pool deck of uniform width is going to be built around a rectangular pool that is 20 feet
long and 15 feet wide. After the deck is built, a total of 414 square feet will be occupied.
How wide is the deck encompassing the pool?
NUCC | Secondary II Math 32
Unit 5.4
Solutions:
Space saver for answers/solutions for unit 5.4
NUCC | Secondary II Math 33
Unit 5.5
5.5 SOLVE BY COMPLETING THE SQUARE
Teacher Notes
Time Frame: 1 class period (90 minutes)
Materials Needed: Pencil
Purpose: Solve quadratic equations by completing the square. Convert the standard form of a
𝑏
quadratic expression: ax2 + bx + c into the form a(x + 2 )2 + k.
Core Standards Focus: A.SSE.3b Choose and produce an equivalent form of an expression to
reveal and explain properties of the quantity represented by the expression. b. Complete the
square in a quadratic expression to reveal the maximum or minimum value of the function it
defines.
Related Standards: Squares and square roots
Launch (Whole Class): Hand out the Garden Space Task. Allow students 10 minutes to
complete individually.
Explore (Individual, small group or pairs):
Assign each student a partner and have them go over their findings from the task assignment.
Discuss (Whole Class or Group):
Allow some students to come up to the board and write their results. Discuss these with the class.
NUCC | Secondary II Math 34
Unit 5.5
Example 1: Make a perfect square trinomial
Find the value of c that makes x2 – 10x + c a perfect square trinomial.
Then write the expression as the square of a binomial.
−10
Step 1: Find half the coefficient of x.
2
= -5
Step 2: Square the result of Step 1.
(-5)2 = 25
Step 3: Replace c with the result of Step 2.
x2 – 10x + 25
The trinomial x2 – 10x + c is a perfect square when c = 25.
So, x2 – 10x + 25 = (x – 5)(x – 5) = (x – 5)2.
Example 2: Solve ax2 + bx + c = 0 when a = 1
Solve x2 – 16x + 8 = 0 by completing the square.
Note: Emphasize the need to write all steps and to keep them organized.)
Solution
x2 – 16 x + 8 = 0
Write original equation.
x2 – 16x = -8
Write left side in the form x2 + bx.
x2 – 16x + 64 = -8 + 64
Add (
(x – 8)2 = 56
Write left side as a binomial squared.
x – 8 = ±√56
Take square roots of each side.
x = 8 ± √56
Solve for x.
x = 8 ± 2√14
Simplify: √56 = √4  √14 = 2√14
−16 2
)
2
= 64 to each side.
The solutions are 8 + 2√14 and 8 - 2√14.
Example 3: Solve ax2 + bx + c = 0 when a ≠ 1
Solve 3x2 + 6x - 15 = 0 by completing the square.
3x2 + 6x - 15 = 0
Write original equation.
x2 + 2x - 5 = 0
Divide each side by the coefficient of x2, 3.
NUCC | Secondary II Math 35
Unit 5.5
x2 + 2x = 5
Write left side in the form x2 + bx.
x2 + 2x + 1 = 5 + 1
Add ( 2)2 = 12 = 1 to each side.
(x + 1)2 = 6
Write left side as a binomial squared.
x + 1 = ± √6
Take square roots of each side.
x = -1 ± √6
Solve for x.
2
The solutions are -1 + √6 and -1 - √6 .
Assignment: Ready, Set, Go!
NUCC | Secondary II Math 36
Unit 5.5
NUCC | Secondary II Math 37
Unit 5.5
Mathematics Content
Cluster Title: Write expressions in equivalent forms to solve problems.
Standard A.SSE.3: Choose and produce an equivalent from of an expression to reveal and
explain properties of the quantity represented by the expression.
a. Factor a quadratic expression to reveal the zeros of the function it defines.
b. Complete the square in a quadratic expression to reveal the maximum or minimum value of
the function it defines.
c. Use the properties of exponents to transform expressions for exponential functions. (For
example the expression 1.15’ can be rewritten as (1.151/2)12t – 1.01212t to reveal the
approximate equivalent monthly interest rate if the annual rate is 15%.)
Concepts and Skills to Master


Rewrite expressions in different forms using mathematical properties.
Given a context determine the best form of an expression.
Critical Background Knowledge


Understand the distributive property in simplifying and expanding expressions.
Various types of factoring skills.
Academic Vocabulary
Factors, coefficients, terms, exponent, base, constant, variable, binomial, monomial, polynomial
Suggested Instructional Strategies


Connect point-slope form to transformation of a line.
Connect to the forms of a quadratic function.
Skills Based Task:
Problem Task:

NCTM Horseshoes in Flight Task:
http://nctm.org/standards/contet.aspx?id=23749



Given a quadratic in standard form,
rewrite in vertex form and list the
properties used in each step.
One of the factors of 0.2x3 -1.2x2 -0.6x is
(x-2). Find the other factors.
Find multiple ways to rewrite x6 – y6.
Rewrite x ? in radical form.
Some Useful Websites:
Illuminations: Proof Without Words: Completing the Square
http://illuminations.nctm.org/ActivityDetail.aspx?ID=132
NUCC | Secondary II Math 38
Unit 5.5
Garden Space
A Develop Understanding Task 5
Name_____________________________________
Hour___________
1. The length of a garden is 6 feet longer than the width, and the area is 35 square feet. An
equation x(x + 6) = 35 can be used to find the width x. Write an equation in standard form.
2. Can the equation in question 1 be solved using factoring? If so, please show.
3. Can the equation in question 1 be solved using the square root method? If so, please show.
4. Simplify the expression (x + 3)2 – 44. How does this expression relate to the garden
problem?
5. Use the expression (x + 3)2 – 44 and a calculator to find the width of the garden.
NUCC | Secondary II Math 39
Unit 5.5
Ready, Set, Go!
Name__________________________________________________
Hour____________
Ready
Find the value of ‘c’ that makes the expression a perfect square trinomial. Then write the
expression as a square of a binomial.
1. x 2  4 x  c
2. x 2  2 x  c
3. x 2  18 x  c
4. x 2  24 x  c
5. x 2  14 x  c
6. x 2  5 x  c
7. x 2  x  c
8. x 2  7 x  c
Set
Simplify the expression.
Solve the equation by completing the square.
9. x 2  2 x  2  0
10. x 2  6 x  3  0
NUCC | Secondary II Math 40
Unit 5.5
11. x 2  8 x  2  0
12. x 2  2 x  5  0
13. x 2  10 x  11  0
14. x 2  14 x  10  0
15. x 2  x  1  0
16. x 2  x  3  0
NUCC | Secondary II Math 41
Unit 5.5
Go!
Solve the equations by completing the square.
17. 2 x 2  16 x  8  0
18. 5 x 2  10 x  30  0
Find the value of x.
19. Area of rectangle = 40
20. Area of rectangle = 78
x
x
x7
x3
NUCC | Secondary II Math 42
Unit 5.5
Solutions:
Space saver for answers/solutions for unit 5.5
NUCC | Secondary II Math 43
Unit 5.6
5.6 SOLVE WITH COMPLEX NUMBERS
Teacher Notes
Time Frame: 1 class period (90 minutes)
Materials Needed: Pencils
Purpose: Students will solve quadratic equations with solutions involving imaginary numbers.
Students will also add, subtract, multiply, and divide complex numbers.
Core Standards Focus: N.CN.7 Solve quadratic equations with real coefficients that have
complex solutions.
Related Standards: Solving quadratic equations using square roots FOIL method.
Launch (Whole Class): Hand out the task worksheet. Allow students 10 minutes to complete
individually.
Explore (Pair/Share):
Assign small groups and allow students to share their answers with each other. Have students
come up with their own ‘answer key’. Read the answers and see which group got the most
correct.
NUCC | Secondary II Math 44
Unit 5.6
Discuss (Whole Class or Group): Have one student from each group come to the board and
share their answers to numbers 12 and/or 13 – showing their work. Go through the problems and
show them there is no solution to these equations, until today! We are going to explore the use of
imaginary numbers.
Vocabulary:
The imaginary unit is defined as i = √−1. This means i2 = -1.
A complex number written in standard form is a number a + bi where a and b are real numbers.
If b ≠ 0, then a + bi is an imaginary number.
Two complex numbers of the form a + bi and a – bi are called complex conjugates.
Example 1: Solve a quadratic equation
Solve 3x2 – 1 = -16
3x2 – 1 = -16
Write original equation.
3x2 = -15
Add 1 to each side.
x2 = -5
Divide each side by 3.
x = ±√−5
Take square roots of each side.
x = ±i√5
Write in terms of i.
The solutions are i√5 and -i√5.
Example 2: Add and subtract complex numbers
Write the expression 9 – (10 + 2i) – 5i as a complex number in standard form.
9 – (10 + 2i) – 5i = (9 – 10 – 2i) – 5i
Definition of complex subtraction.
= (-1 – 2i) – 5i
Simplify.
= -1 – (2 + 5) i
Definition of complex addition.
= -1 – 7i
Write in standard form.
NUCC | Secondary II Math 45
Unit 5.6
Example 3: Multiply and divide complex numbers
Write each expression as a complex number in standard form.
a. (-8 – 3i)(2 + 4i)
b.
5  2i 5  2i 3  8i


3  8i 3  8i 3  8i
=
15  40i  6i  16i 2
9  24i  64i 2

31  46i 31 46i
 
73
73 73
= -16 – 32i – 6i – 12i2
Multiply using FOIL.
= -16 – 38i – 12(-1)
Simplify. Use i2 = -1.
= -16 – 38i + 12
Simplify.
= -4 – 38i
Write in standard form.
Multiply by the conjugate
3−8i
3−8i
to rationalize the denominator.
Multiply using FOIL.
Use i2 = -1. Write in standard form.
NUCC | Secondary II Math 46
Unit 5.6
Assignment: Ready, Set, Go!
NUCC | Secondary II Math 47
Unit 5.6
Mathematics Content
Cluster Title: Use complex numbers in polynomial identities and equations.
Standard N.CN.7: Solve quadratic equations with real coefficients that have complex solutions.
Concepts and Skills to Master
 Understand the meaning of a complex number.
 Solve a quadratic equation.
Critical Background Knowledge
 Understand the meaning of a complex number.
 Solve quadratic equation.
Academic Vocabulary
Complex number, imaginary number, roots, solutions, zeros
Suggested Instructional Strategies
 Connect to quadratic functions that have no x-intercepts.
Skills Based Task:
Problem Task:
Graph and find the solutions to the function
f(x) – (x – 3)2+5. Reflect the parabola across
the line y = 5 at the vertex. Compare and
contrast the graphs and solutions.
Create a quadratic function without x-intercepts
and verify that its solutions are complex.
Some Useful Websites:
NUCC | Secondary II Math 48
Unit 5.6
Review
A Solidify Understanding Task 6
Name_____________________________________
Hour___________
Evaluate the powers.
1. 45
2. (- 2)3
3. (- 2)4
Simplify the expressions.
4. (3 – 2x) + (6 – 3x)
5. (1 + 8y) – (7 + 5y)
6. (15 + 2a) – (9 – 2a)
7. –x (9 – 6x)
8. (2 – b)(5 – 2b)
9. (4 + 3m)2
10. (3 + 7x)(3 – 7x)
11. (2 – 3y)(2 + 3y)
Solve the equations.
12. x2 + 25 = 0
13. 2x2 + 98 = 0
NUCC | Secondary II Math 49
Unit 5.6
Ready, Set, Go!
Name_________________________________________________
Hour____________
Ready
Solve the equation.
1. 7 x 2  13  20
2. x 2  14  2
3. 4 x 2  5  77
4. 3x 2  1  16
Set
Write the expression as a complex number in standard form.
5.
 11  3i    4  6i 
7. 8i  3  i 
9.
7  6i 7  6i 
6. 15   9  4i   7i
8.
 3  5i  4  2i 
10.
1  3i
2i
NUCC | Secondary II Math 50
Unit 5.6
11.
2i
3 i
12.
4  2i
1 i
Solve the equation.
13. x 2  25
14. x 2  49
15. x 2  9  0
16. x 2  9  5
17. x 2  16  20
18. 4 x 2  20  6 x 2  12
Go!
Write the expression as a complex number in standard form.
19.  2  i    3  2i 
20.  3  2i   1  4i 
21.  6  i    3  i 
22. 5  i    3  5i 
NUCC | Secondary II Math 51
Unit 5.6
23. i   5  6i 
24. 2i   2  3i   1  8i 
25. 2i  4  i 
26. 4i  3  2i 
27. 3i  5  3i 
28. 1  i  2  5i 
29.  4  2i  2  3i 
30. 5  3i  4  4i 
31.  3  4i  3  i 
32.  7  i  3  4i 
33.
2
3 i
34.
6
2  3i
35.
1 i
2  2i
36.
2  2i
4  3i
NUCC | Secondary II Math 52
Unit 5.6
Solutions:
Space saver for answers/solutions for unit 5.6
NUCC | Secondary II Math 53
Unit 5.7
5.7 SOLVE USING THE QUADRATIC FORMULA
Teacher Notes
Time Frame: 1 class period (90 minutes)
Materials Needed: Red, green and blue pencils or pens for each student.
Purpose: Students will learn how to solve quadratic equations using the quadratic formula.
Note: The quadratic formula on the graphing calculator will be taught in section 8.
Core Standards Focus: A.REI.4b Solve quadratic equations by inspection, taking square roots,
completing the square, the quadratic formula, and factoring, as appropriate to the initial form of
the equation. Recognize when the quadratic formula gives complex solutions and write them as a
+ bi for real numbers a and b.
Related Standards: Evaluating expressions. Properties of square roots.
Launch (Whole Class): Hand out task worksheet to each student. Have them work on this for
10 minutes.
Explore (Individual, small group or pairs):
Assign partners and have students share their results with each other. How many different ways
can you both find to graph quadratic equations? Describe each way on their paper. Discuss
results for questions #3 and #4.
NUCC | Secondary II Math 54
Unit 5.7
Discuss (Whole Class or Group):
Present these questions to the entire class.
1.
2.
3.
4.
5.
What answers did you and your partner come up with for questions #3 and #4?
What are the different ways we have learned to solve a quadratic equation?
Which one do you like best? Why? Keep a tally on the board of the different ways.
Graphing, finding the square roots, factoring, and completing the square.
Wouldn’t it be easier to solve quadratic equations if there was some kind of formula? We
will learn that formula today!
Review:
Standard form: ax2 + bx + c = 0
What would a, b, and c be in the following equation? 2x2 + 3x + 5 = 0
Vocabulary:
The quadratic formula: Let a, b, and c be real numbers where a ≠ 0.
The solutions of the quadratic equation ax2 + bx + c = 0 are
𝑥=
−𝑏±√𝑏 2 −4𝑎𝑐
2𝑎
Example 1: Solve a quadratic equation with two real solutions.
Solve x2 – 5x = 4.
𝑥=
x=
x=
x2 – 5x = 4
Write original equation.
x2 – 5x – 4 = 0
Write in standard form.
−𝑏±√𝑏 2 −4𝑎𝑐
Quadratic Formula
2𝑎
−(−5)±√(−5)2 −4(1)(−4)
a = 1, b = -5, c = -4
2(1)
5 ± √41
Simplify.
2
The solutions are x =
5+ √41
2
≈ 5.70 and x =
5− √41
2
≈ - 0.70
NUCC | Secondary II Math 55
Unit 5.7
Example 2: Solve a quadratic equation with one real solution.
Solve 4x2 + 10x = -10x – 25.
4x2 + 10x = -10x – 25
Write original equation.
x2 + 20x + 25 = 0
Write in standard form.
𝑥=
x=
x=
−𝑏±√𝑏 2 −4𝑎𝑐
2𝑎
−20 ±√202 −4(4)(25)
2(4)
−20 ± √0
8
x=-
5
2
Quadratic Formula
a = 4, b = 20, c = 25
Simplify.
Simplify.
5
The solution is - 2.
Example 3: Solve a quadratic equation with imaginary solutions
Solve x2 – 6x = -10.
x2 – 6x = -10
Write original equation.
x2 – 6x + 10 = 0
Write in standard form.
x=
x=
x=
6 ±√(−6)2 −4(1)(10)
2(1)
6 ± √−4
2
6 ±2𝑖
2
x=3±i
a = 1, b = -6, c = 10
Simplify.
Rewrite using the imaginary unit i.
Simplify.
The solutions are 3 + i and 3 – i .
NUCC | Secondary II Math 56
Unit 5.7
Assignment: Ready, Set, Go!
NUCC | Secondary II Math 57
Unit 5.7
Mathematics Content
Cluster Title: Solve equations and inequalities in one variable.
Standard A.REI.4b: Solve quadratic equations in one variable.
a. Use the method of completing the square to transform any quadratic equation in x into an
equation of the form (x-p) 2 = q that has the same solutions. Derive the quadratic formula
from this form
b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing
the square, the quadratic formula and factoring, as appropriate to the initial form of the
equation. Recognize when the quadratic formula gives complex solutions and write them as a
 bi for real numbers a and b.
Concepts and Skills to Master
 Complete the square.
 Solve quadratic equations, including complex solutions, using completing the square,
quadratic formula, factoring, and by taking the square root.
 Derive the quadratic formula from completing the square.
 Recognize when one method is more efficient than the other.
 Interpret the discriminant.
 Understand the zero product property and use it to solve a factorable quadratic equation.
Critical Background Knowledge
 Factor
 Simplify radicals
 Understanding of complex numbers (Secondary II: N.CN.1)
 Understand the real number and complex number systems (Secondary II: N.CN.1)
Academic Vocabulary
radicals, complex numbers solve, factor, discriminant
Suggested Instructional Strategies
 Use of algebra tiles to demonstrate simple completing the square problems (see NCTM
MTMS, March 2007, p. 403).
Skills Based Task:
Problem Task:
Solve the equation 6x2 –x –15 = 0 by
factoring and by completing the square.
Justify each method using mathematical
properties.
Solve the quadratic equation 49x2 – 70x +37 = 0
using two methods. Describe the advantages of
each method.
Some Useful Websites:
Illuminations: Proof Without Words: Completing the Square
http://illuminations.nctm.org/ActivityDetail.aspx?ID=132
NUCC | Secondary II Math 58
Unit 5.7
Analyze Graphs of Quadratic Functions
A Develop Understanding Task 7
Name_____________________________________
Hour___________
1. Graph the 3 following quadratic equations on the graph provided.
a. Use red to graph y  x 2  4 x  2
b. Use blue to graph y  x 2  4 x  4
c. Use green to graph y  x 2  4 x  6
2. How many x-intercepts does each equation have?
a. Red?
b. Blue?
c. Green?
3. What effect does changing the value of c have on the graph?
4. How could solutions to the equations be found by graphing?
NUCC | Secondary II Math 59
Unit 5.7
Ready, Set, Go!
Name__________________________________________________
Hour____________
Ready
Write the equation in standard form. Identify a, b, and c.
1. 2x2 + x + 4 = 0
2. x2 – 2x + 3 = 6
3. -3x2 – 2 = x2 + 3x
4. 5x = 2x2 – x + 9
Use the quadratic formula to solve the equation.
5. x2 + 4x = 2
6. 2x2 – 8x = 1
7. 4x2 + 2x = -2x – 1
8. 16x2 – 20x = 4x – 9
9. x2 – 4x + 5 = 0
10. x2 – x = -7
NUCC | Secondary II Math 60
Unit 5.7
Set
Use the quadratic formula to solve the equation.
11. x2 – 3x + 2 = 0
12. x2 + 5x + 2 = 0
13. x2 – 3x + 1 = 0
14. 3x2 + x – 4 = 0
15. 2x2 – 4x – 1 = 0
16. 2x2 – 4x + 1 = 0
17. 3x2 + 2x = 0
18. -2x2 – 2x – 1 = 0
19. 5x2 - 9x + 3 = 0
20. –x2 + 3x – 4 = 2
21. 3x2 + 2x = x2 + x + 1
22. 2x2 – x + 3 = 3x + 7
NUCC | Secondary II Math 61
Unit 5.7
Go!
Find the value of x.
23. Area of rectangle = 17.6
24. Area of parallelogram = 40.5
x
x + 2.3
x
2x
25. Horseshoes A contestant tosses a horseshoe from one pit to another with an initial vertical
velocity of 50 feet per second. The horseshoe is released 3 feet above the ground. Use the
model h = -16t2 + 50t + 3 where h is the height (in feet) and t is the time (in seconds) to tell
how long the horseshoe was in the air.
NUCC | Secondary II Math 62
Unit 5.7
Solutions:
Space saver for answers/solutions for unit 5.7
NUCC | Secondary II Math 63
Unit H5.8
H5.8 FUNDAMENTAL THEOREM OF ALGEBRA
Teacher Notes
Time Frame: 1 class period (90 minutes)
Materials Needed: Graphing Calculator with the Quadratic Formula program installed on it.
(You may need to ask another colleague how to do this or go online to download the program.)
Purpose: Students will learn a way to determine the number of solutions to a quadratic equation
by using the discriminant. Students will also learn how to find the solutions on a graphing
calculator.
Core Standards Focus: N.CN.9 Know the Fundamental Theorem of Algebra; show that it is
true for quadratic polynomials.
Related Standards: Graphing equations on a graphing calculator. Know how to find the degree
of a function.
Launch (Whole Class): Hand out the task worksheet. Allow students 10 minutes to work on the
worksheet.
Explore (Pair/Share): Pair students together and have them compare their answers.
NUCC | Secondary II Math 64
Unit H5.8
Discuss (Whole Class or Group):
Discuss with the whole class. What is the discriminant? What conclusions did you make after
completing the chart? Why is the discriminant important?
Vocabulary:
The Fundamental Theorem of Algebra is used to determine the number of solutions to a
function. Identify the degree of the function which is the maximum number of solutions
of the function.
The discriminant is the part of the quadratic formula that is under the radical sign, and can help
you determine the number of solutions of the quadratic equation ax2 + bx + c = 0. You can use
this as a way to verify your solutions.
b2 – 4ac
Discriminant
Example 1: Use the Fundamental Theorem of Algebra to determine the possible solutions
(zeros) to a quadratic function.
Identify the maximum number of solutions for the function
f(x) = x2 – x – 12
Degree: 2
Max. number of solutions: 2
Solve the equation and verify that the maximum number of solutions is 2.
f(x) = x2 – x – 12
f(x) = (x – 4)(x + 3)
0 = (x – 4)(x + 3)
x–4=0
x+3=0
x=4
and x = -3
Write original function
Factor the function
Set the function equal to 0
Solve each separate equation
Verifies there are 2 solutions to the equation
Example 2: Use the Discriminant
Find the discriminant of the quadratic equation and give the number and type of solutions
of the equation.
a. x2 + 6x + 11
b. x2 + 6x + 9
c. x2 + 6x + 5
Solution
Equation
Discriminant
Solution(s)
ax2 + bx + c = 0
b2 – 4ac
𝑥=
a. x2 + 6x + 11 = 0
b. x2 + 6x + 9 = 0
c. x2 + 6x + 5 = 0
62 – 4(1)(11) = -8
62 – 4(1)(9) = 0
62 – 4(1)(5) = 16
Two imaginary: -3± 𝑖√2
One real: -3
Two real: -5, -1
−𝑏±√𝑏 2 −4𝑎𝑐
2𝑎
NUCC | Secondary II Math 65
Unit H5.8
Example 3: Write quadratic equations when given the solutions
Write the quadratic equation with the following solutions.
a. x = -2, 1
Solution: Work backwards from the answer to the equation.
x = -2
x+2=0
x=1
x–1=0
(x + 2)(x – 1) = 0
x2 + x – 2 = 0
b. x =
9 ±√249
14
Solution:
x=
9+ √249
14
x–(
9+√249
14𝑥
9−√249
14
x=
-
14
14
)=0
=0
14x – 9 - √249 = 0
9−√249
14
9−√249
x–(
14𝑥
14
-
14
)=0
9+√249
14
=0
14x – 9 - √249 = 0
(14x – 9 - √249)(14x – 9 + √249) = 0
196x2 – 126x + 14x√249 – 126x + 81 - 9√249 - 14x√249 + 9√249 – 249 = 0
196x2 – 252x – 168 = 0
Example 4: Solve quadratic equations using a graphing calculator
Note: Please make sure your students have the QUADRAT program on their TI-83 or TI-84
calculators. If you are unfamiliar with this program, please ask another math teacher or go to
http://www.tc3.edu/instruct/sbrown/ti83/quadrat.htm. This website seems like a straight forward
procedure for programming the calculator but I did not try it myself. You could do a search for
other sites if you prefer. I have not searched for Casio or other graphing calculator’s quadratic
programs.
NUCC | Secondary II Math 66
Unit H5.8
To solve with calculator set equation = 0, then switch ‘0’ to ‘y’ and the equation is y = ax2 + bx +
c. Now put a, b, and c into the calculator’s quadratic formula application. Please practice this on
your calculator first so you can answer any questions that the students may have.
Solve x2 + 4x – 2 = 0 using a graphing calculator’s quadratic formula program.
x2 + 4x – 2 = 0
Write original equation.
a = 1, b = 4, c = -2
Use these values to enter into the calculator.
Press ENTER to see the solutions
Press ENTER again to see the discriminant (b2 – 4ac)
Using a graphing calculator, find the solution(s) and the discriminant of the following
quadratic equations.
Equation
Solution
a. x2 + 4x + 4 = 0
x=-2
0
b. x2 – 5x – 6 = 0
x = -1, 6
49
c. 5x2 + 7x + 6 = 0
x = - 0.7 ±𝑖√71
-71
d. 196x2 – 252x – 168 = 0
x=
9 ±√249
14
Discriminant
= 1.77, -0.484
195,216 or 249
NUCC | Secondary II Math 67
Unit H5.8
Assignment: Ready, Set, Go!
NUCC | Secondary II Math 68
Unit H5.8
Mathematics Content
Cluster Title: Use complex numbers in polynomial identities and equations.
Standard N.CN.H.9: Know the Fundamental Theorem of Algebra; show that it is true for
quadratic polynomials.
Concepts and Skills to Master

Know that the Fundamental Theorem of Algebra guarantees that any quadratic function will
have a solution in the complex number system.
Critical Background Knowledge




Understand number systems.
Solve quadratic equations.
Know the definition of a complex number (Sec II: N.CN.1)
Know the meaning of algebraically closed. (see Introduction to Unit 1.)
Academic Vocabulary
Fundamental Theorem of Algebra, solutions, complex, roots, real number system, complex
number system, algebraically closed, multiplicity
Suggested Instructional Strategies


Relate the types of solutions to the different number system.
Connect to the need of different number systems.
Skills Based Task:
Problem Task:
In the system of integer numbers, explain why Why is it better to solve quadratic equations in
there is no answer to the equation: 3x = 5.
the complex number system rather than in the
In the system of rational numbers, explain
real number system?
why there is no answer to the equation: x
2+5=0
Some Useful Websites:
http://www.tc3.edu/instruct/sbrown/ti83/quadrat.htm.
NUCC | Secondary II Math 69
Unit H5.8
Investigating the Discriminant
A Develop Understanding Task 8
Name_____________________________________
Hour___________
The quadratic formula is used to find solutions to quadratic equations.
𝑥=
−𝑏 ± √𝑏 2 − 4𝑎𝑐
2𝑎
The discriminant is the part of the quadratic formula that is under the radical sign, and
can help you determine the number of solutions of the quadratic equation. You can use
this as a way to verify your solutions.
b2 – 4ac
Discriminant
1. Graph the function on a graphing calculator to determine the number of solutions. (If
students do not know how to graph a quadratic function on their calculator, you may need to
quickly demonstrate this for them.) Then calculate the value of the discriminant to complete
the chart.
Function
Number of
solutions
a.
y  x 2  11x  24
Value of
discriminant
b2  4ac 
b.
y  2 x2  4 x  1
b2  4ac 
c.
y  x2  4 x  4
b2  4ac 
d.
y  2 x 2  8x  8
b2  4ac 
e.
y  4 x2  2 x  5
b2  4ac 
f.
y  3x 2  4 x  6
b2  4ac 
2. How does the discriminant help you determine the number of solutions of a quadratic
equation?
NUCC | Secondary II Math 70
Unit H5.8
3. What do you notice about the number of solutions and the value of the discriminant in parts a
and b?
4. What do you notice about the number of solutions and the value of the discriminant in parts c
and d?
5. What do you notice about the number of solutions and the value of the discriminant in parts e
and f?
6. Why do you think the value of the discriminant helps you determine the number of solutions?
Hint: Notice that the discriminant is under a radical sign in the quadratic formula.
NUCC | Secondary II Math 71
Unit H5.8
Ready, Set, Go!
Name__________________________________________________
Hour____________
Ready
Identify the maximum number of solutions to each equation. Solve the equation to verify.
1. x2 – 3x + 2 = 0
2. x2 – 10x + 25 = 0
Find the discriminant (by hand) of the quadratic equation and give the number and type of
solutions of the equation.
3. x2 – 2x – 1 = 0
4. x2 – 12x + 36 = 0
5. x2 + 7x + 14 = 0
6. 2x2 + 3x + 2 = 0
7. 3x2 + 2x – 1 = 0
8. 2x2 - 4x + 5 = 0
Set
Find the discriminant (by hand) and use it to determine if the solution has one real, two real, or
two imaginary solution(s).
9. x2 – 3x + 2 = 0
10. x2 - 2x + 1 = 0
11. x2 + 2x + 5 = 0
12. 2x2 + 3x + 1 = 0
NUCC | Secondary II Math 72
Unit H5.8
13. –x2 – 4x – 6 = 0
14. x2 – 5x – 6 = 0
15. -2x2 + x + 4 = 0
16. 5x2 + 7x + 6 = 0
17. x2 + 4x + 1 = 0
18. –x2 + 3x – 4 = 2
19. 2x2 – 1 = 3x + 4
20. x2 – 4x = -3x + 2
Write the quadratic equation when given the solutions.
21. x = 4, 1
22. x = -5, -2
NUCC | Secondary II Math 73
Unit H5.8
23. x = 7
24. x =
−3 ± √361
16
Solve the quadratic equations using the QUADRAT program on a graphing calculator.
25. y = x2 – x – 12
26. y = 3x2 + 5x + 2
27. y = -2x2 + 4x
28. y = 4x2 – 2x + 5
29. -3x2 – 4x – 6 = 0
30. -2x2 + 8x – 8 = 0
Go!
A geyser sends a blast of boiling water high into the air. During the eruption, the height h (in
feet) of the water t seconds after being forced out from the ground can be modeled by h = -16t2
+ 70t.
31. What is the initial velocity of the boiling water?
32. How long is the boiling water in the air?
NUCC | Secondary II Math 74
Unit H5.8
Solutions:
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NUCC | Secondary II Math 75
Unit 5.9
5.9 OPTIONAL CATAPULT LESSON
Teacher Notes
Time Frame: 2 class periods (180 minutes)
Materials Needed: Timer; computer with internet access; screen game ‘Angry Birds’
downloaded on your computer; and various household items for catapults that will be made later.
Optional: Prizes for contest winners.
Purpose: Students will identify parabolas in the real world. Students will demonstrate how
catapults are developed using quadratic functions.
Core Standards Focus: A.REI.4 Solve quadratic equations by inspection, taking square roots,
completing the square, the quadratic formula, and factoring, as appropriate to the initial form of
the equation. Recognize when the quadratic formula gives complex solutions and write them as a
+ bi for real numbers a and b.
Related Standards: Graphing parabolas, finding zeros of quadratic functions
Launch (Whole Class): Hand out task worksheet. Allow 5 minutes for questions 1-3. Ask a
student for the answer to 3 (parabola). For question 4 set the timer for 5 minutes and have the
students individually write down as many real world examples of parabolas as they can. You
may want them to turn their paper over to answer question 4.
Explore (Pair/Share): Put students in groups of 4 and designate one of the members to be the
“scribe.” Have the scribe make 2 lists: 1. List of the answers that are listed more than once. 2.
List of the unique answers that only 1 person has listed.
NUCC | Secondary II Math 76
Unit 5.9
Discuss (Whole Class or Group):
Write the 2 lists on the board. Have one student from each group come up and enter their
answers in the correct list. Award the student and team with the most unique answers (For
example: shooting a basketball, swinging on a swing, angry birds). Discuss the parabola shapes
of the items mentioned: up, down, wide, and narrow.
Continue to discuss parabola shapes and ask if they have played the game “Angry Birds.” If
possible, demonstrate the game on the computer for the entire class to view. Ask students to
explain how the game works and how points are earned. Say: It would be difficult for all of us to
play angry birds together so we can simulate this by making “catapults.”
Ask: What is a catapult? What is it used for? Go to ‘spaghettiboxkids’ website and show the
different catapults. Class assignment is to make a catapult. Prizes can be awarded for: highest
peak of launch, longest launch, closest to a target drawn on the board.
Angry Birds: http://elevatedmath.com/blog/2011/09/07/angry-birds-can-teach-math/ You
could also Google: Free angry birds online game and decide which site to download (Please do
this before you give this lesson.)
Angry Animals (This is like angry birds): http://hoodamath.com/games/angryanimals.php
Catapults: http://spaghettiboxkids.com/blog/catapult-designs-for-kids/ This website lists a
variety of catapults made from common household items. Students could be assigned a group
and a specific kind of catapult to make it class or at home for a class contest, if desired.
NUCC | Secondary II Math 77
Unit 5.9
Assignment: Ready, Set, Go!
NUCC | Secondary II Math 78
Unit 5.9
Mathematics Content
Cluster Title: Solve equations and inequalities in one variable.
Standard A.REI.4: Solve quadratic equations in one variable.
c. Use the method of completing the square to transform any quadratic equation in x into an
equation of the form (x-p) 2 = q that has the same solutions. Derive the quadratic formula
from this form
d. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing
the square, the quadratic formula and factoring, as appropriate to the initial form of the
equation. Recognize when the quadratic formula gives complex solutions and write them as a
 bi for real numbers a and b.
Concepts and Skills to Master
 Complete the square.
 Solve quadratic equations, including complex solutions, using completing the square,
quadratic formula, factoring, and by taking the square root.
 Derive the quadratic formula from completing the square.
 Recognize when one method is more efficient than the other.
 Interpret the discriminant.
 Understand the zero product property and use it to solve a factorable quadratic equation.
Critical Background Knowledge
 Factor
 Simplify radicals
 Understanding of complex numbers (Secondary II: N.CN.1)
 Understand the real number and complex number systems (Secondary II: N.CN.1)
Academic Vocabulary
radicals, complex numbers solve, factor, discriminant
Suggested Instructional Strategies
 Use of algebra tiles to demonstrate simple completing the square problems (see NCTM
MTMS, March 2007, p. 403).
Skills Based Task:
Solve the equation 6x2 –x –15 = 0 by
factoring and by completing the square.
Justify each method using mathematical
properties.
Problem Task:
Solve the quadratic equation 49x2 – 70x +37 = 0
using two methods. Describe the advantages of
each method.
Some Useful Websites:
http://elevatedmath.com/blog/2011/09/07/angry-birds-can-teach-math/
http://hoodamath.com/games/angryanimals.php
http://spaghettiboxkids.com/blog/catapult-designs-for-kids/
NUCC | Secondary II Math 79
Unit 5.9
Shape of Quadratic Function
A Practice Understanding Task 9
Name_____________________________________
Hour___________
1. Draw the shapes of 3 different quadratic functions.
2. Using the 3 drawings above, describe what they could be a diagram of in the real world.
3. What is the shape called of the graph of a quadratic function?
NUCC | Secondary II Math 80
Unit 5.9
Ready, Set, Go!
Name_________________________________________________
Hour____________
Ready
1. Look thru the website: http://spaghettiboxkids.com/blog/catapult-designs-for-kids/ and
decide which catapult your group would like to make.
____________________________________
Catapult type
Set
2. Describe your group’s plan of how to make the catapult.
Go!
3. Describe the results of your group’s catapult experience!
NUCC | Secondary II Math 81
Unit 5.9
Solutions:
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NUCC | Secondary II Math 82