Quadratics are everywhere in A-level maths, from solving equations to...
Maths29 views·Updated Jun 4, 2026·2 pagesUnderstanding Quadratics: Maths Notes and Examples
Los@los
1
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Solving Quadratic Equations and Functions
You've got several methods at your disposal for tackling quadratic equations, and knowing when to use each one will save you loads of time in exams. Factorising is often the quickest route when the numbers work out nicely, like turning x² - 2x - 15 = 0 into = 0, giving you x = -3 or x = 5.
When factorising doesn't work smoothly, completing the square becomes your best mate. The key formula is x² + bx = ² - ². For example, if you need to solve x² + 8x + 10 = 0, you'd rewrite it as ² - 16 + 10 = 0, which simplifies to ² = 6, so x = -4 ± √6.
Functions add another layer to quadratics that's actually quite straightforward once you get the hang of it. The domain is all possible x-values you can input, whilst the range covers all possible outputs. Roots are simply the x-values where f(x) = 0 - these are the points where your graph crosses the x-axis.
Quick tip: When finding where two functions intersect, set them equal to each other and solve the resulting equation. This technique appears frequently in exam questions!
2
of 2
Quadratic Graphs and the Discriminant
Completing the square isn't just for solving equations - it's brilliant for finding turning points on graphs too. When you get your quadratic into the form f(x) = a² + q, the turning point sits at . For instance, y = x² - 5x + 4 becomes y = ² - 9/4, revealing a minimum point at (5/2, -9/4).
The discriminant is like a crystal ball that tells you exactly what to expect from your quadratic before you even solve it. When b² - 4ac > 0, you'll get two distinct real roots, meaning your parabola crosses the x-axis twice.
If b² - 4ac = 0, you've got exactly one repeated root - your parabola just touches the x-axis at its turning point. When b² - 4ac < 0, there are no real roots at all, so your parabola doesn't touch the x-axis.
Exam strategy: Always check the discriminant first when asked about the number of solutions. It'll tell you immediately what you're dealing with and can save you from lengthy calculations!
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Maths29 views·Updated Jun 4, 2026·2 pagesUnderstanding Quadratics: Maths Notes and Examples
Los@los
Quadratics are everywhere in A-level maths, from solving equations to sketching graphs. This section covers all the essential techniques you need to master quadratic equations, including factorising, completing the square, and using the quadratic formula.
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Solving Quadratic Equations and Functions
You've got several methods at your disposal for tackling quadratic equations, and knowing when to use each one will save you loads of time in exams. Factorising is often the quickest route when the numbers work out nicely, like turning x² - 2x - 15 = 0 into = 0, giving you x = -3 or x = 5.
When factorising doesn't work smoothly, completing the square becomes your best mate. The key formula is x² + bx = ² - ². For example, if you need to solve x² + 8x + 10 = 0, you'd rewrite it as ² - 16 + 10 = 0, which simplifies to ² = 6, so x = -4 ± √6.
Functions add another layer to quadratics that's actually quite straightforward once you get the hang of it. The domain is all possible x-values you can input, whilst the range covers all possible outputs. Roots are simply the x-values where f(x) = 0 - these are the points where your graph crosses the x-axis.
Quick tip: When finding where two functions intersect, set them equal to each other and solve the resulting equation. This technique appears frequently in exam questions!
2
of 2Sign up to see the content. It's free!
- Access to all documents
- Improve your grades
- Join milions of students
Quadratic Graphs and the Discriminant
Completing the square isn't just for solving equations - it's brilliant for finding turning points on graphs too. When you get your quadratic into the form f(x) = a² + q, the turning point sits at . For instance, y = x² - 5x + 4 becomes y = ² - 9/4, revealing a minimum point at (5/2, -9/4).
The discriminant is like a crystal ball that tells you exactly what to expect from your quadratic before you even solve it. When b² - 4ac > 0, you'll get two distinct real roots, meaning your parabola crosses the x-axis twice.
If b² - 4ac = 0, you've got exactly one repeated root - your parabola just touches the x-axis at its turning point. When b² - 4ac < 0, there are no real roots at all, so your parabola doesn't touch the x-axis.
Exam strategy: Always check the discriminant first when asked about the number of solutions. It'll tell you immediately what you're dealing with and can save you from lengthy calculations!
We thought you’d never ask...
What is the Knowunity AI companion?
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.
Where can I download the Knowunity app?
You can download the app from Google Play Store and Apple App Store.
Is Knowunity really free of charge?
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
Similar content
Most popular content in Maths
9Most popular content
9Can't find what you're looking for? Explore other subjects.
Students love us — and so will you.
4.6/5App Store
4.7/5Google 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 SiOS 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 KlichAndroid 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.
AnnaiOS user