Edexcel GCSE Foundation Maths - What are Factorising Quadratics?

Introduction

Factorising quadratics is a key skill in GCSE Maths, allowing you to manipulate algebraic expressions and solve equations. This tutorial will guide you through the basics of factorising quadratics, focusing on the most common types you'll encounter in your exams.

What is a Quadratic Expression?

A quadratic expression is an expression with a variable raised to the power of 2, usually written in the form ax² + bx + c, where a, b, and c are constants. Here are some examples:

• x² + 5x + 6
• 2x² - 3x - 2
• -x² + 4

Why Factorise Quadratics?

Factorising quadratics allows us to:

• Solve quadratic equations: By setting the factorised expression equal to zero, we can find the values of x that satisfy the equation.
• Simplify expressions: Factorising can make complex expressions easier to work with.
• Identify roots: The factors of a quadratic expression reveal the x-intercepts of its graph.

Factorising Quadratics: Step-by-Step

The most common method for factorising quadratics is the "grouping method":

1. Find two numbers that add up to 'b' and multiply to 'ac'. Remember 'a', 'b', and 'c' are the coefficients in the quadratic expression ax² + bx + c.

2. Rewrite the middle term ('bx') using the two numbers found in step 1.

3. Group the first two terms and the last two terms.

4. Factor out the greatest common factor (GCF) from each group.

5. The two terms now have a common factor (the brackets) and can be factored again.

Example:

Factorise x² + 5x + 6

1. Find two numbers that add up to 5 and multiply to 6. These numbers are 2 and 3.

2. Rewrite the middle term: x² + 2x + 3x + 6

3. Group the terms: (x² + 2x) + (3x + 6)

4. Factor out the GCF: x(x + 2) + 3(x + 2)

5. Factor out (x + 2): (x + 2)(x + 3)

Therefore, the factorised form of x² + 5x + 6 is (x + 2)(x + 3).

Special Cases

Perfect Square Trinomials:

• x² + 2bx + b² = (x + b)²
• x² - 2bx + b² = (x - b)²

Difference of Squares:

• x² - b² = (x + b)(x - b)

Practice Makes Perfect

To master factorising quadratics, practice is key. Work through various examples, focusing on different types of expressions. Use online resources or textbooks to find exercises and check your solutions. Don't hesitate to ask your teacher or tutor for guidance if you face difficulties.