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422

5 Apr 2023

9 pages

Easy Steps to Solve Quadratic Equations!

S

Safwan Master

@safwanmaster

A comprehensive guide to solving quadratic equations step by step... Show more

1.
2.
3.
4.
5.
Name:
Score: 51/57
Strengths:
Completing the square
Discriminant
Disguised Quadratics
Solving
Sketching
Any others?
(i)
(ii)

Page 2: Graphical Representations

This page focuses on the visual interpretation of quadratic equations through graphs and diagrams. It presents different forms of quadratic expressions and their corresponding graphical representations.

Example: Four different quadratic curves are shown, demonstrating how different forms of quadratic equations affect their graphical appearance

Vocabulary: Axis of symmetry - the vertical line that divides a parabola into two identical halves

Highlight: The relationship between the algebraic form and graphical representation is emphasized

1.
2.
3.
4.
5.
Name:
Score: 51/57
Strengths:
Completing the square
Discriminant
Disguised Quadratics
Solving
Sketching
Any others?
(i)
(ii)

Page 3: Advanced Quadratic Concepts

The page delves into more complex quadratic equations involving parameters and special conditions like equal roots.

Definition: Equal roots occur when the discriminant of a quadratic equation equals zero

Example: The equation k4k-4x² - 4x + k-2 = 0 is analyzed for equal roots

Highlight: The importance of working with exact values rather than decimal approximations

1.
2.
3.
4.
5.
Name:
Score: 51/57
Strengths:
Completing the square
Discriminant
Disguised Quadratics
Solving
Sketching
Any others?
(i)
(ii)

Page 4: Solving Methods

This section demonstrates various methods for solving quadratic equations step by step.

Example: Detailed solution of x⁴ - 4x² - 5 = 0 using substitution method

Highlight: The importance of verifying solutions and checking for extraneous roots

Definition: Discriminant formula: b² - 4ac determines the nature of roots

1.
2.
3.
4.
5.
Name:
Score: 51/57
Strengths:
Completing the square
Discriminant
Disguised Quadratics
Solving
Sketching
Any others?
(i)
(ii)

Page 5: Applications and Problem Solving

Focuses on practical applications and problem-solving strategies for quadratic equations.

Example: Finding points of intersection between a line and a quadratic curve

Highlight: The relationship between the discriminant and the number of real solutions

Definition: The vertex of a parabola represents either the maximum or minimum point

1.
2.
3.
4.
5.
Name:
Score: 51/57
Strengths:
Completing the square
Discriminant
Disguised Quadratics
Solving
Sketching
Any others?
(i)
(ii)

Page 6: Graphical Analysis

Detailed examination of different forms of quadratic expressions and their graphs.

Example: Analysis of y = 3x3x², y = x² + 9, and y = 3x3xx+3x + 3

Highlight: The relationship between factored form and x-intercepts

Definition: X-intercepts are the points where a curve crosses the x-axis

1.
2.
3.
4.
5.
Name:
Score: 51/57
Strengths:
Completing the square
Discriminant
Disguised Quadratics
Solving
Sketching
Any others?
(i)
(ii)

Page 7: Advanced Problem Solving

Complex problem-solving involving parameters and conditions for real roots.

Example: Finding conditions for no real roots using the discriminant

Highlight: The importance of algebraic manipulation in solving parametric equations

1.
2.
3.
4.
5.
Name:
Score: 51/57
Strengths:
Completing the square
Discriminant
Disguised Quadratics
Solving
Sketching
Any others?
(i)
(ii)

Page 8: Special Cases and Applications

Explores special cases of quadratic equations and their applications.

Example: Solving equations with fractional powers and analyzing their solutions

Highlight: The significance of domain restrictions when solving equations

1.
2.
3.
4.
5.
Name:
Score: 51/57
Strengths:
Completing the square
Discriminant
Disguised Quadratics
Solving
Sketching
Any others?
(i)
(ii)

Page 9: Vertex Form and Applications

Focuses on the vertex form of quadratic equations and its applications.

Example: Finding the vertex form of a quadratic equation given its minimum point

Definition: Vertex form: y = axhx - h² + k, where h,kh,k is the vertex

Highlight: The relationship between the vertex form and the graph's transformations

1.
2.
3.
4.
5.
Name:
Score: 51/57
Strengths:
Completing the square
Discriminant
Disguised Quadratics
Solving
Sketching
Any others?
(i)
(ii)

Page 1: Introduction to Quadratic Equations

This page introduces fundamental concepts for solving quadratic equations and presents various problem types. The questions focus on expressing quadratics in standard forms and finding roots.

Example: Express 2x² + 12x + 23 in the form ax+bx + b² + c

Highlight: The page emphasizes the importance of showing complete working steps when solving equations

Definition: A quadratic equation has repeated roots when its discriminant equals zero



Can't find what you're looking for? Explore other subjects.

Students love us — and so will you.

4.9/5

App Store

4.8/5

Google Play

The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan S

iOS user

This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha Klich

Android user

Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

Anna

iOS user

Best app on earth! no words because it’s too good

Thomas R

iOS user

Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.

Basil

Android user

This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.

David K

iOS user

The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.

Rohan U

Android user

I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.

Xander S

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

iOS user

This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now

Paul T

iOS user

The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan S

iOS user

This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha Klich

Android user

Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

Anna

iOS user

Best app on earth! no words because it’s too good

Thomas R

iOS user

Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.

Basil

Android user

This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.

David K

iOS user

The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.

Rohan U

Android user

I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.

Xander S

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

iOS user

This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now

Paul T

iOS user

 

Maths

422

5 Apr 2023

9 pages

Easy Steps to Solve Quadratic Equations!

S

Safwan Master

@safwanmaster

A comprehensive guide to solving quadratic equations step by step and analyzing their properties.

• The document covers various aspects of quadratic equations including completing the square, finding discriminants, and solving disguised quadratics

• Key topics include expressing quadratic in... Show more

1.
2.
3.
4.
5.
Name:
Score: 51/57
Strengths:
Completing the square
Discriminant
Disguised Quadratics
Solving
Sketching
Any others?
(i)
(ii)

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Page 2: Graphical Representations

This page focuses on the visual interpretation of quadratic equations through graphs and diagrams. It presents different forms of quadratic expressions and their corresponding graphical representations.

Example: Four different quadratic curves are shown, demonstrating how different forms of quadratic equations affect their graphical appearance

Vocabulary: Axis of symmetry - the vertical line that divides a parabola into two identical halves

Highlight: The relationship between the algebraic form and graphical representation is emphasized

1.
2.
3.
4.
5.
Name:
Score: 51/57
Strengths:
Completing the square
Discriminant
Disguised Quadratics
Solving
Sketching
Any others?
(i)
(ii)

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Page 3: Advanced Quadratic Concepts

The page delves into more complex quadratic equations involving parameters and special conditions like equal roots.

Definition: Equal roots occur when the discriminant of a quadratic equation equals zero

Example: The equation k4k-4x² - 4x + k-2 = 0 is analyzed for equal roots

Highlight: The importance of working with exact values rather than decimal approximations

1.
2.
3.
4.
5.
Name:
Score: 51/57
Strengths:
Completing the square
Discriminant
Disguised Quadratics
Solving
Sketching
Any others?
(i)
(ii)

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Page 4: Solving Methods

This section demonstrates various methods for solving quadratic equations step by step.

Example: Detailed solution of x⁴ - 4x² - 5 = 0 using substitution method

Highlight: The importance of verifying solutions and checking for extraneous roots

Definition: Discriminant formula: b² - 4ac determines the nature of roots

1.
2.
3.
4.
5.
Name:
Score: 51/57
Strengths:
Completing the square
Discriminant
Disguised Quadratics
Solving
Sketching
Any others?
(i)
(ii)

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Page 5: Applications and Problem Solving

Focuses on practical applications and problem-solving strategies for quadratic equations.

Example: Finding points of intersection between a line and a quadratic curve

Highlight: The relationship between the discriminant and the number of real solutions

Definition: The vertex of a parabola represents either the maximum or minimum point

1.
2.
3.
4.
5.
Name:
Score: 51/57
Strengths:
Completing the square
Discriminant
Disguised Quadratics
Solving
Sketching
Any others?
(i)
(ii)

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Page 6: Graphical Analysis

Detailed examination of different forms of quadratic expressions and their graphs.

Example: Analysis of y = 3x3x², y = x² + 9, and y = 3x3xx+3x + 3

Highlight: The relationship between factored form and x-intercepts

Definition: X-intercepts are the points where a curve crosses the x-axis

1.
2.
3.
4.
5.
Name:
Score: 51/57
Strengths:
Completing the square
Discriminant
Disguised Quadratics
Solving
Sketching
Any others?
(i)
(ii)

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Page 7: Advanced Problem Solving

Complex problem-solving involving parameters and conditions for real roots.

Example: Finding conditions for no real roots using the discriminant

Highlight: The importance of algebraic manipulation in solving parametric equations

1.
2.
3.
4.
5.
Name:
Score: 51/57
Strengths:
Completing the square
Discriminant
Disguised Quadratics
Solving
Sketching
Any others?
(i)
(ii)

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Page 8: Special Cases and Applications

Explores special cases of quadratic equations and their applications.

Example: Solving equations with fractional powers and analyzing their solutions

Highlight: The significance of domain restrictions when solving equations

1.
2.
3.
4.
5.
Name:
Score: 51/57
Strengths:
Completing the square
Discriminant
Disguised Quadratics
Solving
Sketching
Any others?
(i)
(ii)

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Page 9: Vertex Form and Applications

Focuses on the vertex form of quadratic equations and its applications.

Example: Finding the vertex form of a quadratic equation given its minimum point

Definition: Vertex form: y = axhx - h² + k, where h,kh,k is the vertex

Highlight: The relationship between the vertex form and the graph's transformations

1.
2.
3.
4.
5.
Name:
Score: 51/57
Strengths:
Completing the square
Discriminant
Disguised Quadratics
Solving
Sketching
Any others?
(i)
(ii)

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Page 1: Introduction to Quadratic Equations

This page introduces fundamental concepts for solving quadratic equations and presents various problem types. The questions focus on expressing quadratics in standard forms and finding roots.

Example: Express 2x² + 12x + 23 in the form ax+bx + b² + c

Highlight: The page emphasizes the importance of showing complete working steps when solving equations

Definition: A quadratic equation has repeated roots when its discriminant equals zero

Can't find what you're looking for? Explore other subjects.

Students love us — and so will you.

4.9/5

App Store

4.8/5

Google Play

The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan S

iOS user

This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha Klich

Android user

Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

Anna

iOS user

Best app on earth! no words because it’s too good

Thomas R

iOS user

Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.

Basil

Android user

This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.

David K

iOS user

The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.

Rohan U

Android user

I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.

Xander S

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

iOS user

This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now

Paul T

iOS user

The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan S

iOS user

This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha Klich

Android user

Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

Anna

iOS user

Best app on earth! no words because it’s too good

Thomas R

iOS user

Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.

Basil

Android user

This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.

David K

iOS user

The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.

Rohan U

Android user

I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.

Xander S

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

iOS user

This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now

Paul T

iOS user