Factorising Quadratic Expressions: A Comprehensive Guide

Factorising quadratic expressions is a fundamental and essential skill in algebra. Understanding how to factorise can help in solving equations, simplifying expressions, and conducting various operations in mathematics. We will explore the method of factorising quadratic expressions with detailed steps and examples.

Introduction to Factorisation

Factorisation in mathematics refers to breaking down a complex expression into simpler expressions that can be multiplied together to give the original expression. When it comes to quadratic expressions, we aim to break them down into the product of two binomials. This is crucial because it simplifies the expression and makes it easier to analyze.

Steps to Factorise a Quadratic Expression

1. Identify the form of the quadratic expression: A typical quadratic expression is in the form ax2 bx c, where a, b, and c are coefficients. 2. Find two numbers that multiply to give a*c and add to give b: If ax2 bx c can be factored, then we need to find two numbers that satisfy these conditions. 3. Rewrite the middle term: Split the middle term bx into two terms using the two numbers found.

4. Factor by grouping: Group the terms and factor out the common factors. 5. Check the result: Ensure that the original expression is obtained by expanding the factored form.

Example 1: Factorising 6x5^2 3x^2 × x^1 - 6

Let's walk through the example step-by-step:

Identify the form: 6x5^2 3x^2 x - 6 Find the numbers: We need two numbers that multiply to give 6 * -6 -36 and add to give 1. These numbers are 9 and -4. Rewrite the middle term: 6x5^2 9x - 4x - 6 Factor by grouping: 3x(2x5 3) - 2(2x5 3) Final factorisation: (3x - 2)(x5 3)

Example 2: Factorising 36x^2 6253x^25x2 - 6

Let's break this expression down:

Identification: 36x^2 6253x^25x2 - 6 Grouping and considering the substitution: Let 3x^25x2 y Result after substitution: 12y^26 49y44 Rewriting the middle term and factorising: 12y^233y16y44 or 3y41114y11 Further factorisation: (4y11)(3y4) Returning to the algebraic form: (12x^2211)(9x^215x4)

Example 3: Factorising 123x^25x213x^25x2 - 6

Let's solve this example:

Substitution: Let 3x^25x2 y Expression after substitution: 12y25y2 - 6 Further simplification and factorisation: 12y^249y44 or 3y41114y11 Final step: (4y11)(3y4) Return to the original algebraic form: (12x*2211)(9x^215x4)

Conclusion

Mastering the art of factorisation is crucial for solving a variety of algebraic problems. By following the steps outlined in our examples, you can enhance your ability to simplify expressions, solve equations, and perform various mathematical operations. Continuous practice and understanding the underlying principles will further solidify this skill. Happy factorising!