Factorising quadratic expressions

Factorising Quadratic Expressions

Introduction

  • A Quadratic Expression is an expression of polynomial of degree 2.
  • It generally takes the form: ax² + bx + c where a, b and c are constants.
  • Factorising involves rewriting an expression as the product of its factors.

Key Features of Quadratic Expressions

  • In a quadratic expression, the highest power of the variable (x) is always 2.
  • Quadratic expressions are often useful in representing physical and mathematical situations, such as projectile motion and geometric problems.

Understanding Factorisation of Quadratic Expressions

  1. Begin with an equation that is in the form: ax² + bx + c.

  2. The first step is to look for two numbers that multiply to give ac (product of a and c), and add up to b.

  3. Once these two numbers are identified, rewrite bx as sum (or difference) of these two calculated numbers times x. You should have something like ax² + mx + nx + c.

  4. We now need to ‘group’ the terms in pairs and factor by grouping.

  5. Look for common factor in each pair and ‘remove’ it (meaning place it before an opening bracket). Provide replacements for the removed common factors so that once the bracket is expanded, you will obtain the original quadratic expression.

  6. You should end up with two copies of the same factor. Combine these to complete the factorisation.

Example

  • For example, in the expression 6x² + x - 2, find two numbers that multiply to -12 (product of 6 and -2) and add up to 1. These are 4 and -3.

  • Rewrite the quadratic as: 6x² + 4x - 3x - 2.

  • Group the terms and factor by grouping: 2x(3x + 2) -1(3x + 2)

  • The factorised form (3x + 2)(2x - 1) is obtained.

Conclusion

  • Factorising quadratic expressions can seem complex at first, but with regular practice, the process will become more natural.
  • Always remember to check your factorisation by expanding your answer. If your original quadratic expression is obtained, then the factorisation is correct.
  • Keep in mind factorising is the reverse process of expanding brackets.