Factorising quadratic expressions
Factorising an expression means finding the factors that multiply together to give that expression.
A quadratic expression is one that has an โ๐ยฒโ term as its highest power.
\(\mathbf {x^2}\), \(\mathbf {2x^2 -3x}\), \(\mathbf {x^2 - 9}\) and \(\mathbf {x^2 + 5x + 6}\) are all quadratic expressions.
Some quadratic expressions cannot be factorised.
Factorising quadratic expressions of the form \(\mathbf {x^2 + bx + c}\)
To find a method for factorising an expression such as \(\mathbf {x^2 + 5x + 6}\), look at how that expression was arrived at by expanding two brackets.
There are three terms in the expanded expression:
First term:
๐ยฒ
Second term:
sum of +2๐ and +3๐
Third term:
product of +2 and +3
This information gives us a method for factorising.
Examples
Factorise \(\mathbf {x^2 + 2x โ 15}\):
To Factorise:
- Find two numbers whose sum is +2 and whose product is โ15
The product is minus 15, so one of factors must be negative.
The numbers needed are either:
+5 and -3 or -5 and +3 As the sum is positive, the pair with the higher + value is the one to choose i.e.
+5 and -3
- Write down the factors:
\(\mathbf {x^2 + 2x โ 15 = (x + 5)(x โ 3)}\)
- Answer:
\(\mathbf {x^2 + 2x โ 15 = (x + 5)(x โ 3)}\)
\(\mathbf {(x - 3)(x + 5)}\) is also a correct answer. The order of the factors does not matter.
Question
Factorise \(๐ยฒ + 5๐ โ 24\)
Solution
Identify the product and sum of the two key values that we need to find.
Product = -24
Sum = +5
+8 and -3 add to give +5 and multiply to give -24
The factors are (๐ + 8) and (๐ โ 3)
Answer: \(\mathbf {x^2 + 5x โ 24 = (x + 8)(x โ 3)}\)
E๐ample
Factorise ย ย ย ย ย ย ย ย ย ย ย ย ย ย ย ๐ยฒ - 9๐ + 20
Solution
Identify the product and sum of the two key values that we need to find.
Product = +20
Sum = - 9
-4 and -5 add to give -9 and multiply to give +20The factors are (๐ - 4) and (๐ - 5)
Answer: ๐ยฒ - 9๐ + 20 = (๐ - 4)(๐ - 5)
Question
Factorise xยฒ - 17x + 70
Identify the product and sum of the two key values that we need to find.
Product = +70
Sum = - 17
- -7 and -10 add to give -17 and multiply to give +70
The factors are (๐-7) and (๐-10)
Answer:
๐ยฒ - 17๐ + 70 = (๐-7)(๐-10)
Factorising expressions of the form ๐ยฒ-aยฒ (difference of two squares)
Expressions such as ๐ยฒ-aยฒ can be factorised using the difference of two squares method.
To understand how this works, look at the result when (๐ + 5)(๐ โ 5) is expanded.
(๐ + 5)(๐ โ 5) = ๐(๐ -5) + 5(๐ โ 5) = ๐ยฒ โ 5๐ + 5๐ โ 25 Since = ๐ยฒโ 25 Expanding (๐ + 5)(๐ โ 5) gives ๐ยฒ โ 25
The inverse of this means that ๐ยฒ โ 25 factorises to give (๐ + 5)(๐ โ 5)
- Note that in the expression ๐ยฒ โ 25 ๐ is squared
- 25 = 5ยฒ and there is a minus sign in between so we have the difference of two squares!
In general, ๐ยฒ โ aยฒ can be factorised to give (๐ + a)(๐ โ a)
Both ๐ยฒ and 100 (10ยฒ) are squares and there is a - sign in between.
Use the difference of two squares method - DOTS.
The factors can be written down without any further working.
๐ยฒ โ 100 = ๐ยฒ โ 10ยฒ
= (๐ + 10)(๐ โ 10)
Question
Factorise ๐ยฒ - 49
Solution
๐ยฒ - 49 = ๐ยฒ - 7ยฒ
Use DOTS
Answer
๐ยฒ - 49 = (๐ + 7)(๐ - 7)
Example
Factorise 9 - ๐ยฒ
DOTS can still be used here โ the expression does not have to start with โ๐ยฒโ
9 - ๐ยฒ = 3ยฒ - ๐ยฒ
Factors are (3 + ๐)(3 โ ๐)
Answer:
9 - ๐ยฒ = (3 + ๐)(3 โ ๐)
Difference of two squares (DOTS) often appears on exams
Test yourself
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