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22 Dec 2025

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2 pages

Understanding Quadratics: Maths Notes and Examples

Los

@los

Quadratics are everywhere in A-level maths, from solving equations to... Show more

2) QUADRATICS

2.1 SOLVING QUADRATIC EQUATIONS

* e.g. $x²-2x-15=0$
* $(x+3)(x-5)=0$
* $x=-3, x=5$

* e.g. $(x-3)²=7$
* $x-3=±7$
*

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 = $x + b/2$ ² - $b/2$ ². For example, if you need to solve x² + 8x + 10 = 0, you'd rewrite it as $x + 4$ ² - 16 + 10 = 0, which simplifies to $x + 4$ 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!

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) = a $x + p$ ² + q, the turning point sits at $-p, q$ . For instance, y = x² - 5x + 4 becomes y = $x - 5/2$ ² - 9/4, revealing a minimum point at (5/2, -9/4).

The discriminant $b² - 4ac$ 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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Maths

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22 Dec 2025

•

2 pages

Understanding 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

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!

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Quadratic Graphs and the Discriminant

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!