Factorising Quadratic Expressions
Factorising is the process of rewriting a quadratic expression as a product of two linear brackets. This is a foundational skill in algebra, essential for solving equations and simplifying more complex expressions without a calculator.
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- To factorise x² + bx + c, find two numbers that multiply to make the constant term 'c' and add to make the x-coefficient 'b'. These numbers, p and q, will form the brackets (x + p)(x + q).
- For harder quadratics of the form ax² + bx + c, find two numbers that multiply to a*c and sum to b. Use these to split the middle 'bx' term, then factorise the resulting four-term expression by grouping.
- The 'difference of two squares' is a special case: A² - B² always factorises to (A - B)(A + B). 'A' and 'B' can be numbers, variables, or a mix, like in 9x² - 16 = (3x)² - 4².
- Always check for a common numerical or variable factor across all terms before you begin factorising the quadratic part.
- Be aware that not all quadratic expressions can be factorised using whole numbers.
Formulae
A2 - B2 = (A - B)(A + B) Use this identity when you have an expression with two terms, both of which are perfect squares, with a subtraction sign between them.
Definitions
- Quadratic Expression
- An algebraic expression where the highest power of the variable is 2, generally written in the form ax² + bx + c.
- Factorising
- The process of expressing a polynomial as a product of its factors, which are typically simpler expressions (like linear brackets).
Worked example
Factorise the expression 6x² - 11x - 10 completely.
- 1
This is in the form ax² + bx + c, with a=6, b=-11, c=-10.
There are no common factors to take out first.
- 2
Calculate a*c:
6 × (-10) = -60 - 3
Find two numbers that multiply to -60 and add to -11.
By listing factor pairs of 60, we find -15 and +4 satisfy these conditions (-15 × 4 = -60 and -15 + 4 = -11).
- 4
Split the middle term (-11x) using these numbers:
6x² - 15x + 4x - 10.
- 5
Factorise the expression by grouping the first two terms and the last two terms:
3x(2x - 5) + 2(2x - 5).
- 6
Notice the common bracket (2x - 5).
Factor this out:
(3x + 2)(2x - 5).
Answer: (3x + 2)(2x - 5)
Common mistakes
- ×Sign errors are the most frequent mistake. Be very careful when finding two numbers that multiply to a negative constant 'c' but sum to a positive or negative 'b'.
- ×Forgetting to include an initial common factor in the final answer. For example, factorising 4x² - 4 to (2x-2)(2x+2) instead of the complete factorisation 4(x-1)(x+1).
- ×Confusing the sum and product rules, for example finding two numbers that add to 'c' and multiply to 'b'.
- ×Incorrectly applying the difference of two squares to a sum, such as trying to factorise x² + 25.
No-calculator tips
- ✓When finding factor pairs for a*c, systematically list them to avoid missing the correct pair. Start with 1, 2, 3, etc. and check their corresponding partners.
- ✓After factorising, perform a quick mental expansion of your brackets to check it matches the original expression. The 'first' terms should match ax², the 'last' terms should match 'c', and the 'inner' plus 'outer' should match 'bx'.
- ✓If the 'c' term is positive, the signs in both brackets will be the same (either both + or both -). If 'c' is negative, the signs will be different (one + and one -).