Master Quadratic Equations: Factorising Explained

Understanding Quadratic Equations

Quadratic equations are a key topic for GCSE and A-Level Maths students. They typically take the form:

ax² + bx + c = 0

Where a, b, and c are constants, and 'x' represents the variable. Factorising is one of the most common methods to solve quadratic equations, so let’s break it down step by step.

What Does It Mean to Factorise?

Factorising involves rewriting a quadratic equation as the product of two binomials. For example:

x² + 5x + 6 = (x + 2)(x + 3)

Once factorised, you can solve the equation by setting each bracket equal to zero.

Steps to Factorise Quadratic Equations

1. Identify coefficients: Locate a, b, and c from the equation.
2. Find two numbers: Think of two numbers that multiply to a × c and add to b.
3. Split the middle term: Rewrite bx using the two numbers found.
4. Group and factor: Group terms and factorise each group.
5. Write as binomials: Combine the factors into two brackets.

Example 1: Simple Quadratic

Factorise: x² + 5x + 6

1. Identify coefficients: a = 1, b = 5, c = 6.
2. Find numbers: Two numbers multiplying to 6 and adding to 5 are 2 and 3.
3. Rewrite: x² + 2x + 3x + 6.
4. Group: (x² + 2x) + (3x + 6).
5. Factorise: x(x + 2) + 3(x + 2).
6. Combine: (x + 2)(x + 3).

Example 2: Complex Quadratic

Factorise: 6x² + 7x + 2

1. Identify coefficients: a = 6, b = 7, c = 2.
2. Find numbers: Two numbers multiplying to 12 (a × c) and adding to 7 are 3 and 4.
3. Rewrite: 6x² + 3x + 4x + 2.
4. Group: (6x² + 3x) + (4x + 2).
5. Factorise: 3x(2x + 1) + 2(2x + 1).
6. Combine: (2x + 1)(3x + 2).

Exam Tips for Factorising Quadratic Equations

• Practice: The more you factorise, the quicker you'll spot number pairs.
• Check: Always expand your brackets to verify your factorisation.
• Time management: In exams, prioritise simpler quadratics first.
• Use the discriminant: For A-Level questions, check if factorisation is possible using b² - 4ac.

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Practice Exercises

Try factorising these quadratic equations:

• x² + 7x + 12
• 2x² + 5x + 3
• 3x² - 11x + 10

Solutions:

• (x + 3)(x + 4)
• (2x + 3)(x + 1)
• (3x - 5)(x - 2)

Why Factorising Matters

Factorising quadratic equations is essential for solving problems in algebra, mechanics, and even real-world scenarios. It's also a foundation for future topics, such as completing the square and quadratic formula.

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