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Britain & the wider world: 1745 -1901
1l the quest for political stability: germany, 1871-1991
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Megan
16/12/2025
Maths
WJEC AS-level Pure Mathematics Quadratics
43
•
16 Dec 2025
•
Megan
@megan_0306
GCSE Mathematics takes complex algebraic concepts and transforms them into... Show more
Quadratics are equations in the form ax²+bx+c=0 where a, b, and c are constants. They can be solved by factorising into the form =0, which gives us solutions x=m and x=n.
When factorising isn't straightforward, we can use the quadratic formula: x = /2a. This formula works for any quadratic equation regardless of complexity.
With inequalities involving quadratics, remember that the graph's shape affects the solution. When a is positive, the parabola opens upward (creating a minimum); when a is negative, it opens downward (creating a maximum).
Quick Tip: When solving quadratic inequalities, always sketch the curve to visualise where the function is positive or negative .
Set notation provides a concise way to describe intervals of values. The notation {x : P} represents all values of x that satisfy condition P. For intervals, we use parentheses () for exclusive bounds and square brackets for inclusive bounds.
For example, x < m or x > n can be written as ∪ (n, ∞), while s ≤ x < t would be [s, t).
The discriminant tells us about the nature of a quadratic equation's solutions:
Remember: The discriminant is your quick diagnostic tool - it reveals everything about a quadratic's solutions without requiring you to solve the equation fully!
When solving quadratics, the discriminant helps determine the number and type of solutions:
A positive discriminant gives two different real roots. For example, x²-4x+3=0 has solutions x=1 and x=3.
A zero discriminant produces one repeated root, often called a double root. The equation can be written in the form ². For example, x²-4x+4=0 gives us ² = 0, so x=2.
A negative discriminant means there are no real roots. The quadratic never crosses the x-axis.
Exam Tip: Questions often ask you to find values of parameters that give specific types of roots. Always use the discriminant conditions to solve these problems!
Completing the square transforms a quadratic into the form a. This technique helps find the vertex of parabolas and sometimes simplifies solving.
For simultaneous equations, we have two main approaches:
Method 1 (Elimination): Manipulate equations to eliminate one variable. For example, with 3x+y=29 and 4x+3y=47, multiply the first equation by 3 to get 9x+3y=87. Subtracting the second equation eliminates y, giving 5x=40, so x=8. Substitute back to find y=5.
Method 2 (Substitution): Rearrange one equation to express one variable in terms of another, then substitute. From 3x+y=29, we get y=29-3x. Substituting into 4x+3y=47 leads to 4x+3=47, which simplifies to x=8 and y=5.
Challenge yourself: Try both methods on the same problem to see which one feels more intuitive for you!
Non-linear simultaneous equations involve at least one equation that isn't a straight line. These typically represent the points where two different curves intersect.
For these problems, substitution (Method 2) is almost always the best approach. For example, to find where y=x²-4x+3 and y=3-x intersect:
The solution points are (0,3) and (3,0), representing the exact coordinates where these two curves meet.
Visual insight: Whenever you solve these problems, sketch the curves to verify your answers make sense geometrically!
The shape of a quadratic function y=ax²+bx+c depends critically on the value of a:
When a is positive, the parabola opens upward, creating a minimum point. This means the function has its lowest value at the vertex, and increases as x moves away in either direction.
When a is negative, the parabola opens downward, creating a maximum point. Here, the function reaches its highest value at the vertex.
The vertex form y=a²+k directly gives the turning point (h,k), making it particularly useful for identifying these critical points on the graph.
Practical application: Many optimization problems in real life use quadratics - finding minimum costs or maximum profits often involves finding the vertex of a quadratic function!
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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English LiteratureApp Store
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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
Megan
@megan_0306
GCSE Mathematics takes complex algebraic concepts and transforms them into practical problem-solving tools. In these notes, we explore quadratics, inequalities, set notation and simultaneous equations - fundamental techniques that help us solve a wide range of mathematical problems.
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Quadratics are equations in the form ax²+bx+c=0 where a, b, and c are constants. They can be solved by factorising into the form =0, which gives us solutions x=m and x=n.
When factorising isn't straightforward, we can use the quadratic formula: x = /2a. This formula works for any quadratic equation regardless of complexity.
With inequalities involving quadratics, remember that the graph's shape affects the solution. When a is positive, the parabola opens upward (creating a minimum); when a is negative, it opens downward (creating a maximum).
Quick Tip: When solving quadratic inequalities, always sketch the curve to visualise where the function is positive or negative .
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Improve your grades
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By signing up you accept Terms of Service and Privacy Policy
Set notation provides a concise way to describe intervals of values. The notation {x : P} represents all values of x that satisfy condition P. For intervals, we use parentheses () for exclusive bounds and square brackets for inclusive bounds.
For example, x < m or x > n can be written as ∪ (n, ∞), while s ≤ x < t would be [s, t).
The discriminant tells us about the nature of a quadratic equation's solutions:
Remember: The discriminant is your quick diagnostic tool - it reveals everything about a quadratic's solutions without requiring you to solve the equation fully!
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Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
When solving quadratics, the discriminant helps determine the number and type of solutions:
A positive discriminant gives two different real roots. For example, x²-4x+3=0 has solutions x=1 and x=3.
A zero discriminant produces one repeated root, often called a double root. The equation can be written in the form ². For example, x²-4x+4=0 gives us ² = 0, so x=2.
A negative discriminant means there are no real roots. The quadratic never crosses the x-axis.
Exam Tip: Questions often ask you to find values of parameters that give specific types of roots. Always use the discriminant conditions to solve these problems!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Completing the square transforms a quadratic into the form a. This technique helps find the vertex of parabolas and sometimes simplifies solving.
For simultaneous equations, we have two main approaches:
Method 1 (Elimination): Manipulate equations to eliminate one variable. For example, with 3x+y=29 and 4x+3y=47, multiply the first equation by 3 to get 9x+3y=87. Subtracting the second equation eliminates y, giving 5x=40, so x=8. Substitute back to find y=5.
Method 2 (Substitution): Rearrange one equation to express one variable in terms of another, then substitute. From 3x+y=29, we get y=29-3x. Substituting into 4x+3y=47 leads to 4x+3=47, which simplifies to x=8 and y=5.
Challenge yourself: Try both methods on the same problem to see which one feels more intuitive for you!
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Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Non-linear simultaneous equations involve at least one equation that isn't a straight line. These typically represent the points where two different curves intersect.
For these problems, substitution (Method 2) is almost always the best approach. For example, to find where y=x²-4x+3 and y=3-x intersect:
The solution points are (0,3) and (3,0), representing the exact coordinates where these two curves meet.
Visual insight: Whenever you solve these problems, sketch the curves to verify your answers make sense geometrically!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
The shape of a quadratic function y=ax²+bx+c depends critically on the value of a:
When a is positive, the parabola opens upward, creating a minimum point. This means the function has its lowest value at the vertex, and increases as x moves away in either direction.
When a is negative, the parabola opens downward, creating a maximum point. Here, the function reaches its highest value at the vertex.
The vertex form y=a²+k directly gives the turning point (h,k), making it particularly useful for identifying these critical points on the graph.
Practical application: Many optimization problems in real life use quadratics - finding minimum costs or maximum profits often involves finding the vertex of a quadratic function!
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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Quotes from every main character
English LiteratureApp Store
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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
Sociology