Lesson 15: Using the Quadratic Formula
Lesson 15
NYS COMMON CORE MATHEMATICS CURRICULUM
M4
ALGEBRA I
Lesson 15: Using the Quadratic Formula Classwork Opening Exercises Solve the following: 1.
4๐ฅ๐ฅ 2 + 5๐ฅ๐ฅ + 3 = 2๐ฅ๐ฅ 2 โ 3๐ฅ๐ฅ
2.
๐๐ 2 โ 14 = 5๐๐
Exercises Solve Exercises 1โ5 using the quadratic formula. 1.
๐ฅ๐ฅ 2 โ 2๐ฅ๐ฅ + 1 = 0
Lesson 15: Date:
Using the Quadratic Formula 11/19/14
ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org
S.78 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 15
M4
ALGEBRA I
2.
3๐๐ 2 + 4๐๐ + 8 = 0
3.
2๐ก๐ก 2 + 7๐ก๐ก โ 4 = 0
4.
๐๐ 2 โ 2๐๐ โ 1 = 0
5.
๐๐2 โ 4 = 3
Lesson 15: Date:
Using the Quadratic Formula 11/19/14
ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org
S.79 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 15
NYS COMMON CORE MATHEMATICS CURRICULUM
M4
ALGEBRA I
For Exercises 6โ9, determine the number of real solutions for each quadratic equation without solving. 6.
๐๐2 + 7๐๐ + 33 = 8 โ 3๐๐
7.
7๐ฅ๐ฅ 2 + 2๐ฅ๐ฅ + 5 = 0
8.
2๐ฆ๐ฆ 2 + 10๐ฆ๐ฆ = ๐ฆ๐ฆ 2 + 4๐ฆ๐ฆ โ 3
9.
4๐ง๐ง 2 + 9 = โ4๐ง๐ง
10. On the line below each graph, state whether the discriminant of each quadratic equation is positive, negative, or equal to zero. Then, identify which graph matches the discriminants below. Graph 1
Graph 2
_____________________
Discriminant A: (โ2)2 โ 4(1)(2) Graph: ______
Lesson 15: Date:
Graph 3
______________________
_______________________
Discriminant B: (โ4)2 โ 4(โ1)(โ4)
Discriminant C: (โ4)2 โ 4(1)(0)
Graph: ______
Graph: ______
Graph 4
______________________
Discriminant D: (โ8)2 โ 4(โ1)(โ13) Graph: ______
Using the Quadratic Formula 11/19/14
ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org
S.80 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 15
NYS COMMON CORE MATHEMATICS CURRICULUM
M4
ALGEBRA I
Lesson Summary You can use the sign of the discriminant, ๐๐ 2 โ 4๐๐๐๐, to determine the number of real solutions to a quadratic equation in the form ๐๐๐ฅ๐ฅ 2 + ๐๐๐๐ + ๐๐ = 0, where ๐๐ โ 0. If the equation has a positive discriminant, there are two real solutions. A negative discriminant yields no real solutions, and a discriminant equal to zero yields only one real solution.
Problem Set Without solving, determine the number of real solutions for each quadratic equation. 1. 2. 3. 4.
๐๐ 2 โ 4๐๐ + 3 = 0
2๐๐2 + 7 = โ4๐๐ + 5
๐ฅ๐ฅ โ 3๐ฅ๐ฅ 2 = 5 + 2๐ฅ๐ฅ โ ๐ฅ๐ฅ 2 4๐๐ + 7 = ๐๐ 2 โ 5๐๐ + 1
Based on the graph of each quadratic function, ๐ฆ๐ฆ = ๐๐(๐ฅ๐ฅ), determine the number of real solutions for each corresponding quadratic equation, ๐๐(๐ฅ๐ฅ) = 0. 5.
6.
7.
8.
Lesson 15: Date:
Using the Quadratic Formula 11/19/14
ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org
S.81 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
M4
ALGEBRA I
Lesson 15: Using the Quadratic Formula Classwork Opening Exercises Solve the following: 1.
4๐ฅ๐ฅ 2 + 5๐ฅ๐ฅ + 3 = 2๐ฅ๐ฅ 2 โ 3๐ฅ๐ฅ
2.
๐๐ 2 โ 14 = 5๐๐
Exercises Solve Exercises 1โ5 using the quadratic formula. 1.
๐ฅ๐ฅ 2 โ 2๐ฅ๐ฅ + 1 = 0
Lesson 15: Date:
Using the Quadratic Formula 11/19/14
ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org
S.78 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 15
M4
ALGEBRA I
2.
3๐๐ 2 + 4๐๐ + 8 = 0
3.
2๐ก๐ก 2 + 7๐ก๐ก โ 4 = 0
4.
๐๐ 2 โ 2๐๐ โ 1 = 0
5.
๐๐2 โ 4 = 3
Lesson 15: Date:
Using the Quadratic Formula 11/19/14
ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org
S.79 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 15
NYS COMMON CORE MATHEMATICS CURRICULUM
M4
ALGEBRA I
For Exercises 6โ9, determine the number of real solutions for each quadratic equation without solving. 6.
๐๐2 + 7๐๐ + 33 = 8 โ 3๐๐
7.
7๐ฅ๐ฅ 2 + 2๐ฅ๐ฅ + 5 = 0
8.
2๐ฆ๐ฆ 2 + 10๐ฆ๐ฆ = ๐ฆ๐ฆ 2 + 4๐ฆ๐ฆ โ 3
9.
4๐ง๐ง 2 + 9 = โ4๐ง๐ง
10. On the line below each graph, state whether the discriminant of each quadratic equation is positive, negative, or equal to zero. Then, identify which graph matches the discriminants below. Graph 1
Graph 2
_____________________
Discriminant A: (โ2)2 โ 4(1)(2) Graph: ______
Lesson 15: Date:
Graph 3
______________________
_______________________
Discriminant B: (โ4)2 โ 4(โ1)(โ4)
Discriminant C: (โ4)2 โ 4(1)(0)
Graph: ______
Graph: ______
Graph 4
______________________
Discriminant D: (โ8)2 โ 4(โ1)(โ13) Graph: ______
Using the Quadratic Formula 11/19/14
ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org
S.80 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 15
NYS COMMON CORE MATHEMATICS CURRICULUM
M4
ALGEBRA I
Lesson Summary You can use the sign of the discriminant, ๐๐ 2 โ 4๐๐๐๐, to determine the number of real solutions to a quadratic equation in the form ๐๐๐ฅ๐ฅ 2 + ๐๐๐๐ + ๐๐ = 0, where ๐๐ โ 0. If the equation has a positive discriminant, there are two real solutions. A negative discriminant yields no real solutions, and a discriminant equal to zero yields only one real solution.
Problem Set Without solving, determine the number of real solutions for each quadratic equation. 1. 2. 3. 4.
๐๐ 2 โ 4๐๐ + 3 = 0
2๐๐2 + 7 = โ4๐๐ + 5
๐ฅ๐ฅ โ 3๐ฅ๐ฅ 2 = 5 + 2๐ฅ๐ฅ โ ๐ฅ๐ฅ 2 4๐๐ + 7 = ๐๐ 2 โ 5๐๐ + 1
Based on the graph of each quadratic function, ๐ฆ๐ฆ = ๐๐(๐ฅ๐ฅ), determine the number of real solutions for each corresponding quadratic equation, ๐๐(๐ฅ๐ฅ) = 0. 5.
6.
7.
8.
Lesson 15: Date:
Using the Quadratic Formula 11/19/14
ยฉ 2014 Common Core, Inc. Some rights reserved. commoncore.org
S.81 This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
