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Algebra 199Updated 24 Oct 20264 pages

Master Quadratics: Factorisation and Solutions - GCSE/IGCSE Guide

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Quadratic expressions might look intimidating at first, but they're actually quite manageable once you know the tricks. Think of factorising as reverse engineering - you're breaking down a complex expression into simpler building blocks that multiply together.
Factorising and Solving Quadratics - GCSE/IGCSE Study Notes – page 1

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Getting Started with Factorising

Factorising quadratics means rewriting expressions like x² - 3x - 28 as two brackets multiplied together. It's like finding the "ingredients" that make up the original expression.

The key is finding two numbers that multiply to give you the constant term (the number without x). For x² - 3x - 28, you need pairs that multiply to make 28: (1,28), (2,14), or (4,7).

Next, check which pair adds or subtracts to give you the middle coefficient. Since 4 - 7 = -3, you get x + 4$$x - 7. Always double-check by expanding your answer!

Quick Tip: Start by listing all factor pairs of the constant term - this saves time and prevents mistakes.

Factorising and Solving Quadratics - GCSE/IGCSE Study Notes – page 2

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Difference of Two Squares (DOTS)

When you spot something like x² - 64, you're looking at a difference of two squares - a special pattern that's much easier to factorise than regular quadratics.

Both terms must be perfect squares x2and82=64x² and 8² = 64 with a minus sign between them. The answer always follows the pattern: x + 8$$x - 8.

This works because when you expand these brackets, the middle terms +8xand−8x+8x and -8x cancel each other out perfectly. No need for trial and error - just take the square root of each term and use opposite signs.

Remember: DOTS always gives you a + b$$a - b where a² and b² are your original terms.

Factorising and Solving Quadratics - GCSE/IGCSE Study Notes – page 3

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Harder Quadratics and Simplifying Fractions

When the coefficient of x² isn't 1 (like in 2x² + 3x + 1), factorising becomes trickier but follows the same logic. Start with factors of the x² term: here it's 2x and x.

For the constant term, find factor pairs (1 and 1 in this case), then test different arrangements in brackets. Expand the middle terms to check: 2x + 1$$x + 1 gives 2x + x = 3x ✓

Factorising helps simplify algebraic fractions too. When you have quadratics in both numerator and denominator, factorise both and cancel common factors. It's like cancelling fractions but with algebra!

Pro Strategy: When arranging factors, try the combination where numbers are closest together first - it often works faster.

Factorising and Solving Quadratics - GCSE/IGCSE Study Notes – page 4

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Solving Quadratic Equations

Once you've factorised a quadratic equation like 15x² + x - 2 = 0, solving becomes straightforward. If two brackets multiply to give zero, one of them must equal zero.

For 15x², try factor pairs like (5x, 3x) rather than (15x, x) - closer numbers usually work better. After some trial with arranging factors, you get 5x + 2$$3x - 1 = 0.

This means either 5x + 2 = 0 or 3x - 1 = 0, giving you x = -2/5 or x = 1/3. Always give both solutions unless the question asks for just one.

Check Yourself: Substitute your answers back into the original equation - both should make it equal zero.

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