Quadratics are everywhere in A-level maths, from solving equations to...
Maths30 views·Updated 4 Sept 2026·2 pagesUnderstanding Quadratics: Maths Notes and Examples
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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 x + 3$$x - 5 = 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 = 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 = a² + q, the turning point sits at . For instance, y = x² - 5x + 4 becomes y = ² - 9/4, revealing a minimum point at .
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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Maths30 views·Updated 4 Sept 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 x + 3$$x - 5 = 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 = 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!
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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 = a² + q, the turning point sits at . For instance, y = x² - 5x + 4 becomes y = ² - 9/4, revealing a minimum point at .
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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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.
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