Quadratics (factor/formula/completing square) - Mathematics IGCSE Study Notes
Overview
Imagine you're throwing a ball, or designing a bridge, or even just trying to figure out how much something grows over time. All these things often follow a special curved path, not a straight line. That curved path is what **quadratics** help us understand! Quadratics are super useful in real life because they describe many natural shapes and movements. Think about the arc of a basketball shot, the shape of satellite dishes, or even how profits change in a business. Learning about quadratics helps us predict and understand these patterns. In these notes, we'll learn different 'superpowers' (methods) to solve problems involving these curved paths: **factorising**, using the **quadratic formula**, and **completing the square**. Each method is like a different tool in your maths toolbox, ready to help you tackle quadratic challenges!
What Is This? (The Simple Version)
Think of a straight line, like the path a car takes on a flat road. We describe that with equations like y = 2x + 1. But what if the path isn't straight? What if it's a curve, like the path of a ball thrown in the air?
That's where quadratics come in! A quadratic is a special type of equation where the highest power of 'x' is 2 (like x²). It always makes a 'U' shape (or an upside-down 'U' shape) when you draw it on a graph. We call this shape a parabola.
Our goal with quadratics is often to find where this 'U' shape crosses the x-axis (the horizontal line). These crossing points are called the roots or solutions of the quadratic equation. Finding them is like finding where the thrown ball hits the ground!
Real-World Example
Imagine you're a cannonball designer (a fun job, right?). You want to know how far your cannonball will travel before it hits the ground. You know the path it takes is a parabola, described by a quadratic equation like: height = -5t² + 20t (where 't' is time after firing).
To find when it hits the ground, you need to know when the 'height' is zero. So, you set the equation to 0: -5t² + 20t = 0. Solving this quadratic equation will tell you the exact time ('t') the cannonball spends in the air before landing. If you solve it, you'll find t=0 (when it starts) and t=4 (when it lands). So, your cannonball flies for 4 seconds!
Method 1: Factorising (The 'Puzzle' Method)
Factorising is like breaking down a big number into its smaller multiplication parts (e.g., 6 = 2 x 3). With quadratics, we break the equation into two simpler brackets that multiply together. 1. Make sure your quadratic is in the form **ax² + bx + c = 0**. (This means all terms are on one side, an...
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Key Concepts
- Quadratic Equation: An equation where the highest power of the variable (usually x) is 2, creating a curved graph.
- Parabola: The 'U' shaped or upside-down 'U' shaped curve that is formed when you graph a quadratic equation.
- Roots (or Solutions): The points where the parabola crosses the x-axis (the horizontal line) on a graph, which are the answers to the quadratic equation.
- Factorising: A method to solve quadratic equations by breaking them down into two simpler brackets that multiply together.
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Exam Tips
- →Always check if the equation is equal to zero before attempting to solve it using any method.
- →If factorising seems too hard or doesn't work quickly, switch to the quadratic formula – it always works!
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