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422
•
5 Apr 2023
•
Easy Steps to Solve Quadratic Equations!
S
Safwan Master
@safwanmaster
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
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 x² - 4x + k-2 = 0 is analyzed for equal roots
Highlight: The importance of working with exact values rather than decimal approximations
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
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
Page 6: Graphical Analysis
Detailed examination of different forms of quadratic expressions and their graphs.
Example: Analysis of y = ², y = x² + 9, and y =
Highlight: The relationship between factored form and x-intercepts
Definition: X-intercepts are the points where a curve crosses the x-axis
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
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
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 = a² + k, where is the vertex
Highlight: The relationship between the vertex form and the graph's transformations
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 a² + 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
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
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
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 x² - 4x + k-2 = 0 is analyzed for equal roots
Highlight: The importance of working with exact values rather than decimal approximations
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
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
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 = ², y = x² + 9, and y =
Highlight: The relationship between factored form and x-intercepts
Definition: X-intercepts are the points where a curve crosses the x-axis
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
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
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 = a² + k, where is the vertex
Highlight: The relationship between the vertex form and the graph's transformations
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 a² + 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