Factorising quadratics

Factorising where a=1

It is possible to factorise a quadratic expression, such as x^2+3x+2, into a factorised form*, such as (x+2)(x+1).

For a simple quadratic where a (the coefficient of x^2) is 1, we find two numbers which multiply to give 2, and add to give 3. In this case: 2 and 1.

Example: factorise x^2+5x+6

We need two numbers which multiply to give 6, and add to give 5.
The numbers 2 and 3 work, since 2\times3=6 and 2+3=5.
Therefore, we can factorise x^2+5x+6 as (x+2)(x+3).

Factorising where a\ne1

When a is not equal to 1, we can use a method called splitting the middle term (or simply called ‘the split method’).

First, we multiply a and c together. Then, we find two numbers which multiply to give ac, and add to give b. We then split the middle term using these two numbers, and factor by grouping.

Example: factorise 2x^2+7x+3

a=2, b=7 and c=3.
ac=2\times3=6.
We need two numbers which multiply to give 6, and add to give 7
The numbers 6 and 1 work, since 6\times1=6 and 6+1=7.
Split the middle term into 6x and 1x:
- 2x^2+6x+1x+3
Use the distributive law to factor by grouping:
- 2x(x+3)+1(x+3)
Factor out the common factor (x+3):
- (x+3)(2x+1)
Answer: (x+3)(2x+1).

Example: factorise 3x^2-11x+10

a=3, b=-11 and c=10.
ac=3\times10=30.
We need two numbers which multiply to give 30, and add to give $-11
The numbers -5 and -6 work, since -5\times-6=30 and -5+(-6)=-11.
Split the middle term into -5x and -6x:
- 3x^2-5x-6x+10
Use the distributive law to factor by grouping:
- x(3x-5)-2(3x-5)
Factor out the common factor (3x-5):
- (3x-5)(x-2)
Answer: (3x-5)(x-2).

Solving quadratic equations by factorising

Please see solving-quadratics-by-factorising.

flashcards

Question	Answer
What are the conditions for the two numbers when factorising x^2 + bx + c?	We need two numbers which multiply to give c, and add to give b.
Factorise x^2 + 5x + 6.	(x+2)(x+3), because 2 \times 3 = 6 and 2 + 3 = 5.
What is the first step in the split method for factorising ax^2 + bx + c when a \neq 1?	Multiply a and c together to get ac.
In the split method, what condition must the two numbers satisfy after finding ac?	The two numbers must multiply to give ac, and add to give b.
Factorise 2x^2 + 7x + 3 using the split method.	1. ac=6, numbers are 6 and 1. 2. Split: 2x^2+6x+1x+3. 3. Group: 2x(x+3)+1(x+3). 4. Factor: (x+3)(2x+1).
Factorise 3x^2 - 11x + 10 using the split method.	1. ac=30, numbers are -5 and -6. 2. Split: 3x^2-5x-6x+10. 3. Group: x(3x-5)-2(3x-5). 4. Factor: (3x-5)(x-2).