Download Set 10 Solving Quadratic Equations by Finding Square Roots

Reasoning with Equations and Inequalities
Set 10: Solving Quadratic Equations by Finding Square Roots
Instruction
Goal: To provide opportunities for students to develop concepts and skills related to solving
quadratic equations using the square root property. The quadratic equations will have two
solutions, one solution, or no real solutions.
Common Core Standards
Algebra: Reasoning with Equations and Inequalities
Solve equations and inequalities in one variable.
A-REI.4.
Solve quadratic equations in one variable.
• 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.
Student Activities Overview and Answer Key
Station 1
Students will be given a number cube. Students will use the number cube to populate quadratic
equations that can be solved by using the square root property. They will realize that quadratic
equations solved by the square root property have answers which are the +/– of the same number.
Answers
1. Answers will vary. Make sure they add the constant to both sides of the equation.
2. Answers will vary. Make sure they take square root of both sides of the equation.
3. Answers will vary. Make sure they find the +/– square root of the constant.
4. Two, because the x2 exponent tells you that there are two solutions.
5. Answers will vary. Make sure that students first add the constant to both sides of the equation,
then take the square root of both sides, then add the constant from the left side to the right side
of the equation.
6. Answers will vary. Make sure that students first add the constant to both sides of the equation,
then take the square root of both sides, then add the constant from the left side to the right side
of the equation.
7. Two solutions, because the square root of a number is the +/– of the number; the square of a
positive is a positive and the square of a negative is also a positive.
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© 2011 Walch Education
Algebra I Station Activities for Common Core Standards
Reasoning with Equations and Inequalities
Set 10: Solving Quadratic Equations by Finding Square Roots
Instruction
Station 2
Students will be given 12 index cards with the following written on them:
a = 1; a = 2; a = 4; a = 5; b = 0; b = 0; b = 0; b = 0; c = 16; c = 25; c = 36; c = 49
Students work together to set up quadratic equations and solve them using the square root property.
Answers
1. Answers will vary. Possible answer: a = 1, b = 0, and c = 16, so
x 2 − 16 = 0
x 2 = 16
x = ±4
2. Answers will vary. Possible answer: a = 2, b = 0, and c = 36, so
2 x 2 − 36 = 0
2 x 2 = 36
x 2 = 18
x = ±3 2
3. Answers will vary. Possible answer: a = 5, b = 0, and c = 25, so
5 x 2 = 25
x2 = 5
x=± 5
4. Answers will vary. Possible answer: a = 4, b = 0, and c = 49, so
4 x 2 = 49
49
x2 =
4
49
7
x=±
=±
4
2
5. two solutions, because you find the square root
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Algebra I Station Activities for Common Core Standards
© 2011 Walch Education
Reasoning with Equations and Inequalities
Set 10: Solving Quadratic Equations by Finding Square Roots
Instruction
Station 3
Students will be given a deck of playing cards that contain only the numbers 2–10. Students will
draw numbers and use them to populate quadratic equations. They will solve the quadratic equations
using the square root property. They will realize that the square root of a negative number yields no
real solutions. They will also realize that the square root of 0 yields only one solution.
Answers
1. Answers will vary. Make sure students subtract the constant from both sides of the equation.
2. negative, because of subtraction
3. You can’t find the square root of a negative number in the real number system.
4. no real solutions
5. Answers will vary. Make sure students subtract the constant from both sides of the equation.
6. It cancels out, so the right-hand side of the equation is 0.
7. x = ± 0 = 0
8. ± 0 = 0
9. 3x2 – 6 = –6
3x2 = 0
x=0
10. 5x2 + 125 = 0
5x2 = –125
x2 = –25
x = no real solutions
Station 4
Students will be given five index cards with the following written on them:
2 x 2 " 72 ; 5 x 2 − 2 x − 32 = 3 x 2 − 2 x + 40 ; x 2 " 36 ; x = ±6 ;
5 x 2 − 3 x 2 − 2 x + 2 x = 32 + 40
Students will arrange the index cards in order to demonstrate how to find the solution to a
quadratic equation that is not given in standard form. Students will indicate their strategy and
acknowledge the square root property. Then they will solve quadratic equations not given in
standard form using the square root property.
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© 2011 Walch Education
Algebra I Station Activities for Common Core Standards
Reasoning with Equations and Inequalities
Set 10: Solving Quadratic Equations by Finding Square Roots
Instruction
Answers
1. 5 x 2 − 2 x − 32 = 3 x 2 − 2 x + 40
2. 5 x 2 − 3 x 2 − 2 x + 2 x = 32 + 40
3. 2 x 2 " 72
4. x 2 " 36
5. x = ±6
6. square root property
7. Answers will vary.
8. 2x2 + 3x = 144 + 3x + x2
x2 = 144
x = ±12
9. 10x2 – 4x + 16 + 4x = 16
10x2 = 0
x=0
10. x2 + 3x = – x2 + 3x – 8
2x2 = –8
x2 = –4
x = no real solutions
Materials List/Setup
Station 1
number cube
Station 2
12 index cards with the following written on them:
a = 1; a = 2; a = 4; a = 5; b = 0; b = 0; b = 0; b = 0; c = 16; c = 25; c = 36; c = 49
Station 3
deck of playing cards that contain only the numbers 2–10
Station 4
five index cards with the following written on them:
2 x 2 " 72 ; 5 x 2 − 2 x − 32 = 3 x 2 − 2 x + 40 ; x 2 " 36 ; x = ±6 ;
5 x 2 − 3 x 2 − 2 x + 2 x = 32 + 40
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Algebra I Station Activities for Common Core Standards
© 2011 Walch Education
Reasoning with Equations and Inequalities
Set 10: Solving Quadratic Equations by Finding Square Roots
Instruction
Discussion Guide
To support students in reflecting on the activities and to gather some formative information about
student learning, use the following prompts to facilitate a class discussion to “debrief” the station
activities.
Prompts/Questions
1. How many solutions does a quadratic equation have?
2. If the standard form of a quadratic equation is ax2 + bx + c = 0, which letter (a, b, or c) should be
equal to 0 in order to use the square root property?
3. If x2 = a negative number, how many solutions does the equation have in the real number system?
4. If x2 = 0, how many solutions does the equation have?
5. Why does the square root property yield answers that are the ± version of the same number?
Think, Pair, Share
Have students jot down their own responses to questions, then discuss with a partner (who was not
in their station group), and then discuss as a whole class.
Suggested Appropriate Responses
1. two solutions
2. b = 0
3. no real solutions
4. one solution, which equals zero
5. Because there are two possible solutions, a positive number can be squared to equal a positive
number, and a negative number can be squared to yield that same positive number. For
example, 3 squared is 9, and so is (–3) squared.
Possible Misunderstandings/Mistakes
• Not realizing that there are two solutions when x2 = a positive number
• Not realizing that there are no real solutions when x2 = a negative number
• Not realizing that there is only one solution when x2 = 0
• Incorrectly combining like terms before solving an equation
225
© 2011 Walch Education
Algebra I Station Activities for Common Core Standards
NAME:
Reasoning with Equations and Inequalities
Set 10: Solving Quadratic Equations by Finding Square Roots
Station 1
At this station, you will find a number cube. Roll the number cube and write the number in the first
box below. Roll it again, and write the second number in the second box.
x2 −
=0
As a group, solve this quadratic equation by finding the value of x.
1. What is the first step you take to solve this equation?
2. What is the second step you take to solve this equation?
3. What is the solution to this equation? __________________
4. How many solutions did you find? Explain your answer.
Given: ( x −
)2 −
=0
Roll the number cube and write the number in the box above. Repeat this process until each box
contains a number.
5. Solve this equation. Show your work.
continued
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Algebra I Station Activities for Common Core Standards
© 2011 Walch Education
NAME:
Reasoning with Equations and Inequalities
Set 10: Solving Quadratic Equations by Finding Square Roots
6. What strategy did you use to solve this equation?
7. How many solutions did you find? Explain your answer.
227
© 2011 Walch Education
Algebra I Station Activities for Common Core Standards
NAME:
Reasoning with Equations and Inequalities
Set 10: Solving Quadratic Equations by Finding Square Roots
Station 2
At this station, you will find 12 index cards with the following written on them:
a = 1; a = 2; a = 4; a = 5; b = 0; b = 0; b = 0; b = 0; c = 16; c = 25; c = 36; c = 49
Place the index cards in three piles, organized by their letter: a, b, and c. For each problem, work
as a group by choosing a card from each pile and writing it in the form shown. For example, if you
draw the cards “a = 4,” “b = 0,” and “c = 16,” inserting them into the form ax2 + bx – c results in this
equation:
4x2 + 0x = 16
Then solve each equation for x using the square root property. Show your work.
1. ax2 + bx – c = 0
2. ax2 + bx – c = 0
continued
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Algebra I Station Activities for Common Core Standards
© 2011 Walch Education
NAME:
Reasoning with Equations and Inequalities
Set 10: Solving Quadratic Equations by Finding Square Roots
3. ax2 + bx = c
4. ax2 + bx = c
5. How many solutions did you find for each problem? Explain your answer.
229
© 2011 Walch Education
Algebra I Station Activities for Common Core Standards
NAME:
Reasoning with Equations and Inequalities
Set 10: Solving Quadratic Equations by Finding Square Roots
Station 3
At this station, you will find a deck of playing cards that contain only the numbers 2–10. Place the
cards in a pile. As a group, select one card. Write the number on the face of the card in the box below.
x2 +
=0
1. What is the first step you take to solve for x?
2. Is the constant now positive or negative? Explain your answer.
3. How does the sign of the constant affect finding the square root of x?
4. What is the solution for x? __________________
Select another card from the pile. Write the number on the face of the card in each box below.
x2 +
=
5. What is the first step you take to solve for x?
6. What happens to the constant?
continued
230
Algebra I Station Activities for Common Core Standards
© 2011 Walch Education
NAME:
Reasoning with Equations and Inequalities
Set 10: Solving Quadratic Equations by Finding Square Roots
7. Based on your new constant, what is the value of x? __________________
8. Why does x only have one value?
Work together to solve problems 9 and 10. Show your work.
9. 3 x 2 − 6 = − 6
10. 5 x 2 + 125 = 0
231
© 2011 Walch Education
Algebra I Station Activities for Common Core Standards
NAME:
Reasoning with Equations and Inequalities
Set 10: Solving Quadratic Equations by Finding Square Roots
Station 4
At this station, you will find five index cards with the following written on them:
2 x 2 " 72 ; 5 x 2 − 2 x − 32 = 3 x 2 + 2 x + 40 ; x 2 " 36 ; x = ±6 ;
5 x 2 − 3 x 2 − 2 x + 2 x = 32 + 40
Shuffle the cards. As a group, arrange the cards to show how to find the solution to
2
5 x − 2 x − 32 = 3 x 2 − 2 x + 40 .
Write the steps on the lines below.
1. __________________
2. __________________
3. __________________
4. __________________
5. __________________
6. What property did you use to find the solutions for x?
7. What strategy did you use to arrange the cards?
As a group, solve problems 8–10. Show your work.
8. 2 x 2 + 3 x = 144 + 3 x + x 2
continued
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Algebra I Station Activities for Common Core Standards
© 2011 Walch Education
NAME:
Reasoning with Equations and Inequalities
Set 10: Solving Quadratic Equations by Finding Square Roots
9. 10 x 2 − 4 x + 16 + 4 x = 16
10. x 2 + 3 x = − x 2 + 3 x − 8
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© 2011 Walch Education
Algebra I Station Activities for Common Core Standards