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 ‘²’ 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:
²
Second term:
sum of +2x and +3x
Third term:
product of +2 and +3
This information gives us a method for factorising.
Examples
Factorise \(\mathbf {x^2 + 2x – 15}\):
To Factorise:
- Find two numbers whose sum is +2 and whose product is –15
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:
\(\mathbf {x^2 + 2x – 15 = (x + 5)(x – 3)}\)
- Answer:
\(\mathbf {x^2 + 2x – 15 = (x + 5)(x – 3)}\)
\(\mathbf {(x - 3)(x + 5)}\) is also a correct answer. The order of the factors does not matter.
Question
Factorise \(² + 5x – 24\)
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 (x + 8) and (x – 3)
Answer: \(\mathbf {x^2 + 5x – 24 = (x + 8)(x – 3)}\)
Example
Factorise ² - 9x + 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 +20
The factors are (x - 4) and (x - 5)
Answer: ² - 9x + 20 = (x - 4)(x - 5)
Question
Factorise ² - 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 (x-7) and (x-10)
Answer:
² - 17x + 70 = (x-7)(x-10)
Factorising expressions of the form ²-a² (difference of two squares)
Expressions such as ²-a² can be factorised using the difference of two squares method.
To understand how this works, look at the result when (x + 5)(x – 5) is expanded.
(x + 5)(x – 5) = x(x -5) + 5(x – 5) = ² – 5x + 5x – 25 Since = ²– 25 Expanding (x + 5)(x – 5) gives ² – 25
The inverse of this means that ² – 25 factorises to give (x + 5)(x – 5)
- Note that in the expression ² – 25 x is squared
- 25 = 5² and there is a minus sign in between so we have the difference of two squares!
In general, ² – a² can be factorised to give (x + a)(x – a)
Both ² and 100 (10²) 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.
² – 100 = ² – 10²
= (x + 10)(x – 10)
Question
Factorise ² - 49
Solution
² - 49 = ² - 72
Use DOTS
Answer
² - 49 = (x + 7)(x - 7)
Example
Factorise 9 - ²
DOTS can still be used here – the expression does not have to start with ‘²”
9 - ² = 3² - ²
Factors are (3 + x)(3 – x)
Answer:
9 - ² = (3 + x)(3 – x)
Difference of two squares (DOTS) often appears on exams
Test yourself
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