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 āšĀ²ā 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:
šĀ²
Second term:
sum of +2š and +3š
Third term:
product of +2 and +3
This information gives us a method for factorising.
Examples
Factorise \(\mathbf {x^2 + 2x ā 15}\):
To Factorise:
- Find two numbers whose sum is +2 and whose product is ā15
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:
\(\mathbf {x^2 + 2x ā 15 = (x + 5)(x ā 3)}\)
- Answer:
\(\mathbf {x^2 + 2x ā 15 = (x + 5)(x ā 3)}\)
\(\mathbf {(x - 3)(x + 5)}\) is also a correct answer. The order of the factors does not matter.
Question
Factorise \(šĀ² + 5š ā 24\)
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 (š + 8) and (š ā 3)
Answer: \(\mathbf {x^2 + 5x ā 24 = (x + 8)(x ā 3)}\)
Ešample
Factorise Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā Ā šĀ² - 9š + 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 (š - 4) and (š - 5)
Answer: šĀ² - 9š + 20 = (š - 4)(š - 5)
Question
Factorise xĀ² - 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 (š-7) and (š-10)
Answer:
šĀ² - 17š + 70 = (š-7)(š-10)
Factorising expressions of the form šĀ²-aĀ² (difference of two squares)
Expressions such as šĀ²-aĀ² can be factorised using the difference of two squares method.
To understand how this works, look at the result when (š + 5)(š ā 5) is expanded.
(š + 5)(š ā 5) = š(š -5) + 5(š ā 5) = šĀ² ā 5š + 5š ā 25 Since = šĀ²ā 25 Expanding (š + 5)(š ā 5) gives šĀ² ā 25
The inverse of this means that šĀ² ā 25 factorises to give (š + 5)(š ā 5)
- Note that in the expression šĀ² ā 25 š is squared
- 25 = 5Ā² and there is a minus sign in between so we have the difference of two squares!
In general, šĀ² ā aĀ² can be factorised to give (š + a)(š ā a)
Both šĀ² and 100 (10Ā²) 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.
šĀ² ā 100 = šĀ² ā 10Ā²
= (š + 10)(š ā 10)
Question
Factorise šĀ² - 49
Solution
šĀ² - 49 = šĀ² - 7Ā²
Use DOTS
Answer
šĀ² - 49 = (š + 7)(š - 7)
Example
Factorise 9 - šĀ²
DOTS can still be used here ā the expression does not have to start with āšĀ²ā
9 - šĀ² = 3Ā² - šĀ²
Factors are (3 + š)(3 ā š)
Answer:
9 - šĀ² = (3 + š)(3 ā š)
Difference of two squares (DOTS) often appears on exams
Test yourself
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