1.3 Factorising
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Learning Objectives
- Factorise quadratic expressions
- Factorise expressions with common factors
- Factorise by grouping
- Solve quadratic equations by factorising
Key Concepts
Introduction to Factorising
Factorising is the reverse process of expanding brackets. It involves expressing an algebraic expression as a product of factors.
Factorising helps simplify expressions and is an essential technique for solving equations and inequalities.
Factorising Quadratic Expressions
To factorise a quadratic expression of the form ax² + bx + c, we look for two numbers that multiply to give ac and add to give b.
This method is commonly known as 'splitting the middle term'.
Example 1
Factorise x² + 5x + 6
Solution:
We look for two numbers that multiply to 6 (the constant term) and add to 5 (the coefficient of x). These numbers are 2 and 3, so we can factorise as (x + 2)(x + 3).
Example 2
Factorise 2x² + 7x + 6
Solution:
We look for two numbers that multiply to 12 (2 × 6) and add to 7. These numbers are 3 and 4, so we factorise as 2x² + 3x + 4x + 6 = x(2x + 3) + 2(2x + 3) = (x + 2)(2x + 3).
Factorising Expressions with Common Factors
If all terms in an expression have a common factor, we can factor it out.
For example, in the expression 4x + 8, both terms share a common factor of 4, so we can factor it out as 4(x + 2).
Example 1
Factorise 3x + 6
Solution:
Both terms have a common factor of 3, so we factor it out as 3(x + 2).
Example 2
Factorise 5x² + 10x
Solution:
Both terms have a common factor of 5x, so we factor it out as 5x(x + 2).
Factorising by Grouping
When an expression has four terms, we can factorise it by grouping. This involves grouping terms in pairs and then factorising each pair separately.
After factoring both pairs, we look for a common factor in both groups.
Example 1
Factorise x² + 5x + 2x + 10
Solution:
Group terms: (x² + 5x) + (2x + 10). Factor each group: x(x + 5) + 2(x + 5). Now factor out the common factor: (x + 5)(x + 2).
Example 2
Factorise 3x² + 9x + 2x + 6
Solution:
Group terms: (3x² + 9x) + (2x + 6). Factor each group: 3x(x + 3) + 2(x + 3). Now factor out the common factor: (x + 3)(3x + 2).
Factorising Quadratic Equations
Quadratic equations can often be solved by factorising the expression into two factors and setting each factor equal to zero.
This is a useful method for solving equations of the form ax² + bx + c = 0.
Example 1
Solve x² + 5x + 6 = 0
Solution:
Factorise the quadratic: (x + 2)(x + 3) = 0. Now set each factor equal to zero: x + 2 = 0 or x + 3 = 0. So, x = -2 or x = -3.
Example 2
Solve x² - 4x = 0
Solution:
Factorise the quadratic: x(x - 4) = 0. Now set each factor equal to zero: x = 0 or x - 4 = 0. So, x = 0 or x = 4.
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