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305
•
22 Dec 2025
•
Bethyn King
@bethyn_k.04
This concise guide covers essential mathematics topics including transformations of... Show more
Line-circle intersections are determined by solving their equations simultaneously. When b² - 4ac > 0, a line intersects a circle twice; when b² - 4ac = 0, the line is tangent to the circle; and when b² - 4ac < 0, there are no intersections.
Tangents and normals are essential concepts for circles. Remember that a tangent to a circle is perpendicular to the radius at the point of contact. If you need to find the length of a tangent from an external point to a circle, use the Pythagoras theorem with the distance from the point to the centre and the radius.
When working with binomial expansions, use factorial notation where n! means n factorial, and the binomial coefficient formula ⁿCᵣ = n!/. This helps you expand expressions like ⁿ using the binomial theorem.
Remember: In any circle problem, converting to the standard form ² + ² = r² helps you immediately identify the centre (a,b) and radius r!
The binomial expansion of ⁿ follows the pattern: 1 + nx + x² + x³ + .... When expanding expressions like ⁷, you can use this formula to quickly find the coefficients of each term.
Function transformations change how graphs look on the coordinate plane. When you add a constant to a function, like f(x) + a, the graph shifts vertically - upward for positive values, downward for negative values. For f, the graph moves horizontally - left for positive values, right for negative values.
With af(x), you're stretching the function vertically by factor a, and if a is negative, you're also reflecting it across the x-axis. For f(ax), the function stretches horizontally by a factor of 1/a, with negative values causing reflection across the y-axis.
In coordinate geometry, remember these essential formulas: gradient formula m = /, line equation y - y₁ = m, midpoint formula /2, /2), and distance formula √.
Pro tip: For circle equations, remember the standard form ² + ² = r², where (a,b) is the centre and r is the radius. This makes finding intersection points with lines much easier!
When working with circles, use the equation x² + y² = r² for circles centered at the origin, and ² + ² = r² for circles with center (a,b). You can expand this to the general form x² + y² - 2ax - 2by + a² + b² - r² = 0 when needed.
The remainder theorem states that when a polynomial f(x) is divided by , the remainder equals f(a). This powerful shortcut helps you find remainders without long division. If f(a) = 0, then is a factor of f(x).
To determine if is a factor, evaluate f. For instance, when checking if is a factor of 3x² + 11x - 8, calculate f(-5). If the result isn't zero, it's not a factor. Similarly, for divisions by expressions like , substitute x = 1/2 into f(x) to find the remainder.
When factorising polynomials, try evaluating f(x) at common values like -2, -1, 0, 1, 2 until you find a value that gives zero. This indicates a factor. For example, if f(1/2) = 0, then is a factor. You can then use polynomial division to find the remaining factors.
Exam tip: In polynomial problems, when you're told that is a factor, it means f(a) = 0, which gives you an equation to help find unknown coefficients!
For complete factorisation of polynomials like 4x³ + 8x² - 11x - 15, identify one factor first , then use polynomial division to find the quadratic factor, and finally factorise the quadratic. This helps you sketch the graph by identifying where it crosses the x-axis (the roots of the polynomial).
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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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
Bethyn King
@bethyn_k.04
This concise guide covers essential mathematics topics including transformations of functions, coordinate geometry, and algebraic techniques. These concepts are crucial for your mathematics coursework and exams, giving you the tools to tackle both graphical and algebraic problems with confidence.
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Line-circle intersections are determined by solving their equations simultaneously. When b² - 4ac > 0, a line intersects a circle twice; when b² - 4ac = 0, the line is tangent to the circle; and when b² - 4ac < 0, there are no intersections.
Tangents and normals are essential concepts for circles. Remember that a tangent to a circle is perpendicular to the radius at the point of contact. If you need to find the length of a tangent from an external point to a circle, use the Pythagoras theorem with the distance from the point to the centre and the radius.
When working with binomial expansions, use factorial notation where n! means n factorial, and the binomial coefficient formula ⁿCᵣ = n!/. This helps you expand expressions like ⁿ using the binomial theorem.
Remember: In any circle problem, converting to the standard form ² + ² = r² helps you immediately identify the centre (a,b) and radius r!
The binomial expansion of ⁿ follows the pattern: 1 + nx + x² + x³ + .... When expanding expressions like ⁷, you can use this formula to quickly find the coefficients of each term.
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Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Function transformations change how graphs look on the coordinate plane. When you add a constant to a function, like f(x) + a, the graph shifts vertically - upward for positive values, downward for negative values. For f, the graph moves horizontally - left for positive values, right for negative values.
With af(x), you're stretching the function vertically by factor a, and if a is negative, you're also reflecting it across the x-axis. For f(ax), the function stretches horizontally by a factor of 1/a, with negative values causing reflection across the y-axis.
In coordinate geometry, remember these essential formulas: gradient formula m = /, line equation y - y₁ = m, midpoint formula /2, /2), and distance formula √.
Pro tip: For circle equations, remember the standard form ² + ² = r², where (a,b) is the centre and r is the radius. This makes finding intersection points with lines much easier!
When working with circles, use the equation x² + y² = r² for circles centered at the origin, and ² + ² = r² for circles with center (a,b). You can expand this to the general form x² + y² - 2ax - 2by + a² + b² - r² = 0 when needed.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
The remainder theorem states that when a polynomial f(x) is divided by , the remainder equals f(a). This powerful shortcut helps you find remainders without long division. If f(a) = 0, then is a factor of f(x).
To determine if is a factor, evaluate f. For instance, when checking if is a factor of 3x² + 11x - 8, calculate f(-5). If the result isn't zero, it's not a factor. Similarly, for divisions by expressions like , substitute x = 1/2 into f(x) to find the remainder.
When factorising polynomials, try evaluating f(x) at common values like -2, -1, 0, 1, 2 until you find a value that gives zero. This indicates a factor. For example, if f(1/2) = 0, then is a factor. You can then use polynomial division to find the remaining factors.
Exam tip: In polynomial problems, when you're told that is a factor, it means f(a) = 0, which gives you an equation to help find unknown coefficients!
For complete factorisation of polynomials like 4x³ + 8x² - 11x - 15, identify one factor first , then use polynomial division to find the quadratic factor, and finally factorise the quadratic. This helps you sketch the graph by identifying where it crosses the x-axis (the roots of the polynomial).
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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Explore the Quadratic Formula and Discriminant to solve quadratic equations and analyze their roots. Learn methods for solving simultaneous equations, including elimination and substitution, and understand how to tackle linear and quadratic inequalities. This summary provides essential techniques and examples for mastering these key algebra concepts.
MathsExplore key concepts of cubic and quartic functions, including reflections, stretches, and translations. This resource covers essential definitions and transformations of graphs, making it ideal for Year 12 Edexcel A Level Maths revision.
MathsExplore key concepts in trigonometry and algebra, including exact trigonometric values, index laws, linear inequalities, surds, and sequences. This summary provides essential insights for mastering these mathematical principles, ideal for exam preparation and revision.
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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
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