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514
•
11 Feb 2026
•
Safwan Master
@safwanmaster
Quadratic equations, modelling, and simultaneous equations are fundamental tools in... Show more
This test focuses on seven key quadratic skills that form the backbone of A-level algebra. Completing the square transforms expressions like $5x^2 + 10x + 25^2 - 3$, making it easier to find turning points and solve equations.
The discriminant $b^2 - 4ac$ tells you about the nature of roots. When it equals zero, you get equal roots - this is crucial for finding when equations have exactly one solution. For example, has equal roots when .
Real-world modelling appears through projectile motion problems. The arrow's trajectory shows how quadratics describe parabolic paths, where the constant 1.8 represents the initial height above ground.
Key Tip: Always check your completed square form by expanding it back out - this catches sign errors that can cost you marks.
Simultaneous equations involving quadratics create intersection problems between curves and lines. When solving and , you're finding where a parabola meets a straight line. The number of solutions tells you how many intersection points exist.
Tangent conditions occur when the discriminant equals zero. For a line to touch (be tangent to) a curve, there must be exactly one intersection point. This gives you the condition to find the value of .
Projectile modelling extends to golf balls and other real situations. The trajectory passing through , , and creates the equation , showing the classic parabolic flight path.
Model limitations are equally important - assuming flat ground, ignoring air resistance, or neglecting spin effects are typical restrictions you should always consider.
Key Tip: When finding tangent conditions, remember that tangency means the discriminant equals zero - one solution only.
Non-linear simultaneous equations like and require substitution techniques. Replace the linear equation into the quadratic to create a solvable form, then work backwards to find both variables.
Disguised quadratics appear in equations like $2x^4 - 15x^2 - 50 = 0a = x^22a^2 - 15a - 50 = 0 = 0$. Remember to convert back to find the original variable.
Sketching quadratic functions requires identifying key features: vertex coordinates, y-intercept, and line of symmetry. For , completing the square gives $13 - 2^2(-1, 13)x = -1$.
Proving positivity uses completed square form. Since for all real , expressions like are always positive because they equal .
Key Tip: For surd form answers, always rationalise denominators and simplify completely - examiners expect exact forms, not decimals.
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.
App 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 Knowunity AI. 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 Knowunity AI. 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
Safwan Master
@safwanmaster
Quadratic equations, modelling, and simultaneous equations are fundamental tools in A-level maths that you'll use constantly in further studies. This test covers everything from completing the square to real-world applications like projectile motion - skills that directly connect mathematical theory... Show more
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Join milions of students
By signing up you accept Terms of Service and Privacy Policy
This test focuses on seven key quadratic skills that form the backbone of A-level algebra. Completing the square transforms expressions like $5x^2 + 10x + 25^2 - 3$, making it easier to find turning points and solve equations.
The discriminant $b^2 - 4ac$ tells you about the nature of roots. When it equals zero, you get equal roots - this is crucial for finding when equations have exactly one solution. For example, has equal roots when .
Real-world modelling appears through projectile motion problems. The arrow's trajectory shows how quadratics describe parabolic paths, where the constant 1.8 represents the initial height above ground.
Key Tip: Always check your completed square form by expanding it back out - this catches sign errors that can cost you marks.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Simultaneous equations involving quadratics create intersection problems between curves and lines. When solving and , you're finding where a parabola meets a straight line. The number of solutions tells you how many intersection points exist.
Tangent conditions occur when the discriminant equals zero. For a line to touch (be tangent to) a curve, there must be exactly one intersection point. This gives you the condition to find the value of .
Projectile modelling extends to golf balls and other real situations. The trajectory passing through , , and creates the equation , showing the classic parabolic flight path.
Model limitations are equally important - assuming flat ground, ignoring air resistance, or neglecting spin effects are typical restrictions you should always consider.
Key Tip: When finding tangent conditions, remember that tangency means the discriminant equals zero - one solution only.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Non-linear simultaneous equations like and require substitution techniques. Replace the linear equation into the quadratic to create a solvable form, then work backwards to find both variables.
Disguised quadratics appear in equations like $2x^4 - 15x^2 - 50 = 0a = x^22a^2 - 15a - 50 = 0 = 0$. Remember to convert back to find the original variable.
Sketching quadratic functions requires identifying key features: vertex coordinates, y-intercept, and line of symmetry. For , completing the square gives $13 - 2^2(-1, 13)x = -1$.
Proving positivity uses completed square form. Since for all real , expressions like are always positive because they equal .
Key Tip: For surd form answers, always rationalise denominators and simplify completely - examiners expect exact forms, not decimals.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
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.
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Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
Explore essential mathematical concepts including polynomial theorems, logarithmic properties, trigonometric functions, and integration techniques. This resource covers everything from solving inequalities to understanding exponential functions, providing a solid foundation for A-level mathematics. Ideal for students aiming for top grades.
MathsExplore key techniques for solving simultaneous equations, including quadratic equations and linear systems. This resource covers various methods, such as substitution and elimination, with detailed examples and solutions. Ideal for Edexcel A Level Maths students preparing for exams.
MathsExplore the three main types of mathematical proofs: Proof by Exhaustion, Proof by Contradiction, and Proof by Deduction. This summary includes examples related to prime numbers, square numbers, and geometric proofs, providing a clear understanding of logical reasoning in mathematics.
MathsExplore a comprehensive set of Edexcel A-Level Maths inequalities problems, including compound inequalities, linear inequalities, and multi-step solutions. This resource provides detailed examples and solutions to enhance your understanding of solving inequalities and graphing techniques.
MathsExplore the detailed solution for optimizing the surface area of a cuboid box with a tunnel, as presented in the SQA Higher Mathematics 2019 exam. This study note covers the derivation of the surface area formula, the application of differentiation to find minimum values, and step-by-step explanations for both parts A and B of the question. Ideal for students preparing for optimization problems in mathematics.
MathsThe first three chapters of A-Level maths Year 1!
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 Knowunity AI. 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 Knowunity AI. 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
English Literature
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