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 βπΒ²β 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:
πΒ²
Second term:
sum of +2π and +3π
Third term:
product of +2 and +3
This information gives us a method for factorising.
Examples
Factorise \(\mathbf {x^2 + 2x β 15}\):
To Factorise:
- Find two numbers whose sum is +2 and whose product is β15
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:
\(\mathbf {x^2 + 2x β 15 = (x + 5)(x β 3)}\)
- Answer:
\(\mathbf {x^2 + 2x β 15 = (x + 5)(x β 3)}\)
\(\mathbf {(x - 3)(x + 5)}\) is also a correct answer. The order of the factors does not matter.
Question
Factorise \(πΒ² + 5π β 24\)
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 (π + 8) and (π β 3)
Answer: \(\mathbf {x^2 + 5x β 24 = (x + 8)(x β 3)}\)
Eπample
Factorise πΒ² - 9π + 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 (π - 4) and (π - 5)
Answer:πΒ² - 9π + 20 = (π - 4)(π - 5)
Question
Factorise xΒ² - 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 (π-7) and (π-10)
Answer:
πΒ² - 17π + 70 = (π-7)(π-10)
Factorising expressions of the form πΒ²-aΒ² (difference of two squares)
Expressions such as πΒ²-aΒ² can be factorised using the difference of two squares method.
To understand how this works, look at the result when (π + 5)(π β 5) is expanded.
(π + 5)(π β 5) = π(π -5) + 5(π β 5)= πΒ² β 5π + 5π β 25 Since = πΒ²β 25 Expanding (π + 5)(π β 5) gives πΒ² β 25
The inverse of this means that πΒ² β 25 factorises to give (π + 5)(π β 5)
- Note that in the expression πΒ² β 25 π is squared
- 25 = 5Β² and there is a minus sign in between so we have the difference of two squares!
In general, πΒ² β aΒ² can be factorised to give (π + a)(π β a)
Both πΒ² and 100 (10Β²) 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.
πΒ² β 100 = πΒ² β 10Β²
= (π + 10)(π β 10)
Question
Factorise πΒ² - 49
Solution
πΒ² - 49 = πΒ² - 7Β²
Use DOTS
Answer
πΒ² - 49 = (π + 7)(π - 7)
Example
Factorise 9 - πΒ²
DOTS can still be used here β the expression does not have to start with βπΒ²β
9 - πΒ² = 3Β² - πΒ²
Factors are (3 + π)(3 β π)
Answer:
9 - πΒ² = (3 + π)(3 β π)
Difference of two squares (DOTS) often appears on exams
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