Maths
Chemistry
Biology
English Language
History
Responding to change (a2 only)
Infection and response
Homeostasis and response
Energy transfers (a2 only)
Cell biology
Organisms respond to changes in their internal and external environments (a-level only)
Biological molecules
Organisation
Substance exchange
Bioenergetics
Genetic information & variation
Inheritance, variation and evolution
Genetics & ecosystems (a2 only)
Ecology
Cells
Show all topics
Britain & the wider world: 1745 -1901
1l the quest for political stability: germany, 1871-1991
The cold war
Inter-war germany
Medieval period: 1066 -1509
2d religious conflict and the church in england, c1529-c1570
2o democracy and nazism: germany, 1918-1945
1f industrialisation and the people: britain, c1783-1885
1c the tudors: england, 1485-1603
2m wars and welfare: britain in transition, 1906-1957
World war two & the holocaust
2n revolution and dictatorship: russia, 1917-1953
2s the making of modern britain, 1951-2007
World war one
Britain: 1509 -1745
Show all topics
Maths
20 Dec 2025
1,896
3 pages
A comprehensive guide to completing the square method for quadratic equations and finding turning points in quadratic graphs... Show more
This page explores more complex scenarios in factorising and completing the square in quadratics, particularly when dealing with coefficients other than 1. The example works through 2x² + 10x + 8.
Example Converting 2x² + 10x + 8 into 2² - 4.5
Highlight When the x² coefficient isn't 1, factor it out before completing the square.
Definition The turning point coordinates can be extracted from the completed square form, with x-coordinate being the negative of the number inside the brackets.
This page applies the completing the square method to find the turning point of the quadratic function y = 2x² + 16x + 26, demonstrating practical applications in graphing.
Example Converting 2x² + 16x + 26 into 2² - 6
Highlight The turning point (-4, -6) is essential for sketching the quadratic graph accurately.
Definition The turning point represents the vertex of the parabola, where the function changes from increasing to decreasing or vice versa.
This page demonstrates the fundamental process of completing the square method for quadratic equations using a straightforward example with x² + 8x + 12. The method systematically transforms the quadratic expression into a perfect square form.
Example Converting x² + 8x + 12 into ² - 4
Definition Completing the square involves rewriting a quadratic expression in the form ² + q, where p and q are constants.
Highlight The turning point can be directly read from the completed square form as (-4, -4).
Vocabulary The term "perfect square" refers to an expression that can be written as a binomial squared.
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
You can download the app from Google Play Store and Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
28
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
Master the techniques for solving quadratic equations through factorization. This study note covers key concepts such as factor pairs, factored form, and step-by-step solutions for various quadratic equations. Ideal for students looking to enhance their understanding of algebraic methods. Type: Summary.
MathsExplore key concepts in algebraic expressions and quadratic functions, including the Quadratic Formula, vertex form, and rationalizing surds. This summary provides essential insights for AS Level Year 1 Pure Maths, focusing on simplifying radicals, understanding discriminants, and solving quadratic equations.
MathsExplore the fundamentals of algebra, including variables, expressions, and equations. This summary covers key concepts such as algebraic identities and manipulation techniques, providing clear examples like expanding (2x + 3y)^2. Ideal for students seeking to strengthen their understanding of algebraic principles.
MathsMaster the art of simplifying algebraic expressions through detailed examples and step-by-step solutions. This resource includes practice questions focused on combining like terms and polynomial simplification, ensuring a comprehensive understanding of algebraic manipulation.
MathsMaster key algebra concepts for the GCSE higher tier with this comprehensive overview. Topics include forming and simplifying algebraic expressions, solving linear and quadratic equations, and graphing techniques. Ideal for quick revision and exam preparation.
MathsExplore essential algebraic vocabulary including expressions, equations, formulas, identities, and functions. This summary provides clear definitions and examples to enhance your understanding of algebra concepts. Ideal for students preparing for exams or seeking to strengthen their math skills.
MathsApp Store
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
A comprehensive guide to completing the square method for quadratic equations and finding turning points in quadratic graphs.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
This page explores more complex scenarios in factorising and completing the square in quadratics, particularly when dealing with coefficients other than 1. The example works through 2x² + 10x + 8.
Example: Converting 2x² + 10x + 8 into 2² - 4.5
Highlight: When the x² coefficient isn't 1, factor it out before completing the square.
Definition: The turning point coordinates can be extracted from the completed square form, with x-coordinate being the negative of the number inside the brackets.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
This page applies the completing the square method to find the turning point of the quadratic function y = 2x² + 16x + 26, demonstrating practical applications in graphing.
Example: Converting 2x² + 16x + 26 into 2² - 6
Highlight: The turning point (-4, -6) is essential for sketching the quadratic graph accurately.
Definition: The turning point represents the vertex of the parabola, where the function changes from increasing to decreasing or vice versa.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
This page demonstrates the fundamental process of completing the square method for quadratic equations using a straightforward example with x² + 8x + 12. The method systematically transforms the quadratic expression into a perfect square form.
Example: Converting x² + 8x + 12 into ² - 4
Definition: Completing the square involves rewriting a quadratic expression in the form ² + q, where p and q are constants.
Highlight: The turning point can be directly read from the completed square form as (-4, -4).
Vocabulary: The term "perfect square" refers to an expression that can be written as a binomial squared.
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
You can download the app from Google Play Store and Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
28
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
Master the techniques for solving quadratic equations through factorization. This study note covers key concepts such as factor pairs, factored form, and step-by-step solutions for various quadratic equations. Ideal for students looking to enhance their understanding of algebraic methods. Type: Summary.
MathsExplore key concepts in algebraic expressions and quadratic functions, including the Quadratic Formula, vertex form, and rationalizing surds. This summary provides essential insights for AS Level Year 1 Pure Maths, focusing on simplifying radicals, understanding discriminants, and solving quadratic equations.
MathsExplore the fundamentals of algebra, including variables, expressions, and equations. This summary covers key concepts such as algebraic identities and manipulation techniques, providing clear examples like expanding (2x + 3y)^2. Ideal for students seeking to strengthen their understanding of algebraic principles.
MathsMaster the art of simplifying algebraic expressions through detailed examples and step-by-step solutions. This resource includes practice questions focused on combining like terms and polynomial simplification, ensuring a comprehensive understanding of algebraic manipulation.
MathsMaster key algebra concepts for the GCSE higher tier with this comprehensive overview. Topics include forming and simplifying algebraic expressions, solving linear and quadratic equations, and graphing techniques. Ideal for quick revision and exam preparation.
MathsExplore essential algebraic vocabulary including expressions, equations, formulas, identities, and functions. This summary provides clear definitions and examples to enhance your understanding of algebra concepts. Ideal for students preparing for exams or seeking to strengthen their math skills.
MathsApp Store
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
Sociology