I have added to the original presentation in response to one of the comments.... the result of 'x' is correct on slide 7, take a look at the new version of this ppt to clear up any confusion about why...
This will help you on how to solve quadratic equations by factoring.
For more instructional resources, CLICK me here!
https://tinyurl.com/y9muob6q
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https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u
I have added to the original presentation in response to one of the comments.... the result of 'x' is correct on slide 7, take a look at the new version of this ppt to clear up any confusion about why...
This will help you on how to solve quadratic equations by factoring.
For more instructional resources, CLICK me here!
https://tinyurl.com/y9muob6q
LIKE and FOLLOW me here!
https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u
For more instructional resources, CLICK me here!
https://tinyurl.com/y9muob6q
LIKE and FOLLOW me here!
https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u
For more instructional resources, CLICK me here!
https://tinyurl.com/y9muob6q
LIKE and FOLLOW me here!
https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u
College algebra in context 5th edition harshbarger solutions manualAnnuzzi19
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College algebra real mathematics real people 7th edition larson solutions manualJohnstonTBL
College Algebra Real Mathematics Real People 7th Edition Larson Solutions Manual
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The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
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Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
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This is a presentation by Dada Robert in a Your Skill Boost masterclass organised by the Excellence Foundation for South Sudan (EFSS) on Saturday, the 25th and Sunday, the 26th of May 2024.
He discussed the concept of quality improvement, emphasizing its applicability to various aspects of life, including personal, project, and program improvements. He defined quality as doing the right thing at the right time in the right way to achieve the best possible results and discussed the concept of the "gap" between what we know and what we do, and how this gap represents the areas we need to improve. He explained the scientific approach to quality improvement, which involves systematic performance analysis, testing and learning, and implementing change ideas. He also highlighted the importance of client focus and a team approach to quality improvement.
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http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
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Quadratic equations ppt
1. Many quadratic equations can not be solved by factoring.
Other techniques are required to solve them.
8.1 – Solving Quadratic Equations
x2
= 20 5x2
+ 55 = 0
Examples:
( x + 2)2
= 18 ( 3x – 1)2
= –4
x2
+ 8x = 1 2x2
– 2x + 7 = 0
2
2 5 0x x− − = 44 2
−−= xx
2. If b is a real number and if a2
= b, then a = ±√¯‾.
20
8.1 – Solving Quadratic Equations
Square Root Property
b
x2
= 20
x = ±√‾‾
x = ±√‾‾‾‾4·5
x = ± 2√‾5 –11
5x2
+ 55 = 0
x = ±√‾‾‾
5x2
= –55
x2
= –11
x = ± i√‾‾‾11
3. If b is a real number and if a2
= b, then a = ±√¯‾.
18
8.1 – Solving Quadratic Equations
Square Root Property
b
( x + 2)2
= 18
x + 2 = ±√‾‾
x + 2 = ±√‾‾‾‾9·2
x +2 = ± 3√‾2
x = –2 ± 3√‾2
–4
( 3x – 1)2
= –4
3x – 1 = ±√‾‾
3x – 1 = ± 2i
3x = 1 ± 2i
3
21 i
x
±
=
ix
3
2
3
1
±=
9. The quadratic formula is used to solve any quadratic equation.
2
4
2
x
cb b a
a
− ± −
=
The quadratic formula is:
Standard form of a quadratic equation is:
2
0x xba c+ + =
8.2 – Solving Quadratic Equations
The Quadratic Formula
10. 2
4
2
x
cb b a
a
− ± −
=
8.2 – Solving Quadratic Equations
The Quadratic Formula
02
=++ cbxax
cbxax −=+2
a
c
x
a
b
x
a
a −
=+2
a
c
x
a
b
x
−
=+2
a
b
a
b
22
1
=⋅ 2
22
42 a
b
a
b
=
a
c
a
b
a
b
x
a
b
x −=++ 2
2
2
2
2
44
a
a
a
c
a
b
a
b
x
a
b
x
4
4
44 2
2
2
2
2
⋅−=++
11. 2
4
2
x
cb b a
a
− ± −
=
8.2 – Solving Quadratic Equations
The Quadratic Formula
22
2
2
2
2
4
4
44 a
ac
a
b
a
b
x
a
b
x −=++
2
2
2
2
2
4
4
4 a
acb
a
b
x
a
b
x
−
=++
2
2
2
2
2
4
4
4 a
acb
a
b
x
a
b
x
−
±=++
2
22
4
4
2 a
acb
a
b
x
−
±=
+
2
2
4
4
2 a
acb
a
b
x
−
±=+
a
acb
a
b
x
2
4
2
2
−
±=+
a
acb
a
b
x
2
4
2
2
−
±−=
a
acbb
x
2
42
−±−
=
12. The quadratic formula is used to solve any quadratic equation.
2
4
2
x
cb b a
a
− ± −
=The quadratic formula is:
Standard form of a quadratic equation is: 2
0x xba c+ + =
2
4 8 0x x+ + =
a = 1 c =b = 4 8
2
3 5 6 0x x− + =
a = 3 c =b = 5−
2
2 0x x+ =
a = 2 c =b = 1 0
2
10x = −
a = 1 c =b = 0 106
2
10 0x + =
8.2 – Solving Quadratic Equations
The Quadratic Formula
13. 2
4
2
x
cb b a
a
− ± −
=2
0x xba c+ + =
2
3 2 0x x− + =
2x =1x =
( )1x − ( )2x − 0=
1 0x − = 2 0x − =
8.2 – Solving Quadratic Equations
The Quadratic Formula
14. 2
4
2
x
cb b a
a
− ± −
=2
0x xba c+ + =
2
3 2 0x x− + =
a = 1 c =b = 3− 2
( ) ( ) ( ) ( )
( )
2
3 3 1 24
12
x
− ± −−
=
−
3 9 8
2
x
± −
=
3 1
2
x
±
=
3 1
2
x
±
=
3 1
2
x
+
=
3 1
2
x
−
=
4
2
x =
2x =
2
2
x =
1x =3 1
2
x
±
=
8.2 – Solving Quadratic Equations
The Quadratic Formula
15. 2
4
2
x
cb b a
a
− ± −
=2
0x xba c+ + =
2
2 5 0x x− − =
a = 2 c =b = 1− 5−
( ) ( ) ( ) ( )
( )
2
4
22
1 521
x
−
=
− −±−−
1 1 40
4
x
± +
=
1 41
4
x
±
=
8.2 – Solving Quadratic Equations
The Quadratic Formula
16. 2
4
2
x
cb b a
a
− ± −
=
8.2 – Solving Quadratic Equations
The Quadratic Formula
44 2
−−= xx
044 2
=++ xx
( ) ( )( )
( )42
44411
2
−±−
=x
8
6411 −±−
=x
8
631 −±−
=x
8
631 i
x
±−
=
8
391 ⋅±−
=
i
x
8
731 i
x
±−
= ix
8
73
8
1
±−=
17. 2
4
2
x
cb b a
a
− ± −
=
8.2 – Solving Quadratic Equations
The Quadratic Formula and the Discriminate
The discriminate is the radicand portion of the quadratic
formula (b2
– 4ac).
It is used to discriminate among the possible number and type
of solutions a quadratic equation will have.
b2
– 4ac Name and Type of Solution
Positive
Zero
Negative
Two real solutions
One real solutions
Two complex, non-real
solutions
18. 2
4
2
x
cb b a
a
− ± −
=
8.2 – Solving Quadratic Equations
The Quadratic Formula and the Discriminate
( ) ( )( )2143
2
−−
89 −
b2
– 4ac Name and Type of Solution
Positive
Zero
Negative
Two real solutions
One real solutions
Two complex, non-real
solutions
2
3 2 0x x− + =
a = 1 c =b = 3− 2
1
Positive
Two real solutions
2x = 1x =
19. 2
4
2
x
cb b a
a
− ± −
=
8.2 – Solving Quadratic Equations
The Quadratic Formula and the Discriminate
( ) ( )( )4441
2
−
641−
b2
– 4ac Name and Type of Solution
Positive
Zero
Negative
Two real solutions
One real solutions
Two complex, non-real
solutions
a = c =b =
63−
Negative
Two complex, non-real solutions
044 2
=++ xx
4 1 4
ix
8
73
8
1
±−=
20. 2
4
2
x
cb b a
a
− ± −
=
8.2 – Solving Quadratic Equations
The Quadratic Formula
Given the diagram below, approximate to the nearest foot how many feet
of walking distance a person saves by cutting across the lawn instead of
walking on the sidewalk.
20 feet
x + 2
x
21. 2
4
2
x
cb b a
a
− ± −
=
8.2 – Solving Quadratic Equations
The Quadratic Formula
Given the diagram below, approximate to the nearest foot how many feet
of walking distance a person saves by cutting across the lawn instead of
walking on the sidewalk.
20 feet
x + 2
x
The Pythagorean Theorem
a2
+ b2
= c2
(x + 2)2
+ x2
= 202
x2
+ 4x + 4 + x2
= 400
2x2
+ 4x + 4 = 400
2x2
+ 4x – 369 = 0
2(x2
+ 2x – 198) = 0
22. 2
4
2
x
cb b a
a
− ± −
=
8.2 – Solving Quadratic Equations
The Quadratic Formula
Given the diagram below, approximate to the nearest foot how many feet
of walking distance a person saves by cutting across the lawn instead of
walking on the sidewalk.
20 feet
x + 2
x
The Pythagorean Theorem
a2
+ b2
= c2
2(x2
+ 2x – 198) = 0
( ) ( )( )
( )12
1981422
2
−−±−
=x
2
79242 +±−
=x
2
7962 ±−
=x
23. 2
4
2
x
cb b a
a
− ± −
=
8.2 – Solving Quadratic Equations
The Quadratic Formula
Given the diagram below, approximate to the nearest foot how many feet
of walking distance a person saves by cutting across the lawn instead of
walking on the sidewalk.
20 feet
x + 2
x
The Pythagorean Theorem
a2
+ b2
= c2
=
±−
=
2
7962
x =
±−
2
2.282
2
2.282 +−
=x
2
2.282 −−
=x
2
2.26
=x
1.13=x
2
2.30−
=x
1.15−=xft
24. 2
4
2
x
cb b a
a
− ± −
=
8.2 – Solving Quadratic Equations
The Quadratic Formula
Given the diagram below, approximate to the nearest foot how many feet
of walking distance a person saves by cutting across the lawn instead of
walking on the sidewalk.
20 feet
x + 2
x
The Pythagorean Theorem
a2
+ b2
= c2
1.13=x
ft2.28
ft
=++ 21.131.13
28 – 20 = 8 ft