GCSE: Quadratic Functions and
Simplifying Rational Expressions
Dr J Frost (jfrost@tiffin.kingston.sch.uk)
Last modified: 25th August 2013
Factorising Overview
Factorising means :
To turn an expression into a product of factors.
Year 8 Factorisation
2x2 + 4xz
Factorise
So what factors can we
see here?
2x(x+2z)
Year 9 Factorisation
x2 + 3x + 2
Factorise
(x+1)(x+2)
A Level Factorisation
2x3 + 3x2 – 11x – 6
Factorise
(2x+1)(x-2)(x+3)
Factor Challenge
5 + 10x
x – 2xz
x2y – xy2
10xyz – 15x2y
xyz – 2x2yz2
+ x2y2
Exercises
Extension Question:
What integer (whole number) solutions
are there to the equation xy + 3x = 15
1) 2𝑥 − 4 = 2 𝑥 −? 2
2) 𝑥𝑦 + 𝑦 = 𝑦(𝑥 +? 1)
Answer: 𝑥 𝑦 + 3 = 15. So the two
3) 𝑞𝑟 − 2𝑞 = 𝑞 𝑟 −? 2
expressions we’re multiplying can be 1 ×
15, 15 × 1, 3 × 5, 5 × 3, −1 × −15, …
4) 6𝑥 − 3𝑦 = 3(2𝑥 ?− 𝑦)
? 𝑦) of
This gives solutions (𝑥,
5) 𝑥𝑦𝑧 + 𝑦𝑧 = 𝑦𝑧(𝑥 ?+ 1)
𝟏, 𝟏𝟐 , 𝟑, 𝟐 , 𝟓, 𝟎 , 𝟏𝟓, −𝟐 ,
−𝟏, −𝟏𝟖 , −𝟑, −𝟖 , −𝟓, −𝟔 , (−𝟏𝟓, −𝟒)
6) 𝑥 2 𝑦 + 2𝑦𝑧 = 𝑦 𝑥 2 ?+ 2𝑧
7) 𝑥 3 𝑦 + 𝑥𝑦 2 = 𝑥𝑦 𝑥?2 + 𝑦
8) 5𝑞𝑟 + 10𝑟 = 5𝑟 𝑞 ?+ 2
9) 12𝑝𝑤 2 − 8𝑤 2 𝑦 = 4𝑤 2 𝑝?− 2𝑦
? +3
10) 55𝑝3 + 33𝑝2 = 11𝑝2 5𝑝
11) 6𝑝4 + 8𝑝3 + 10𝑝2 = 2𝑝2 (3𝑝2 +
? 4𝑝 + 5)
12) 10𝑥 3 𝑦 2 + 5𝑥 2 𝑦 3 + 15𝑥 2 𝑦 2 = 5𝑥 2 𝑦 2 (2𝑥?+ 𝑦 + 3)
Factorising out an expression
It’s fine to factorise out an entire expression:
𝑥 𝑥+2 −3 𝑥+2
→
? − 3)
(𝑥 + 2)(𝑥
2
𝑥 𝑥+1 +2 𝑥+1
→
𝑥2 + 𝑥 + 2 ? 𝑥 + 1
𝑎 2𝑐 + 1 + 𝑏 2𝑐 + 1
?
→ (𝑎 + 𝑏)(2𝑐
+ 1)
2 2𝑥 − 3 2 + 𝑥 2𝑥 − 3
→ (5𝑥 − 6)(2𝑥 ?− 3)
Harder Factorisation
𝑝𝑟 + 𝑞𝑠 − 𝑝𝑠 − 𝑞𝑟
? − 𝑞)
= (𝑟 − 𝑠)(𝑝
𝑎𝑏 + 𝑎 + 𝑏 + 1
?
= (𝑎 + 1)(𝑏
+ 1)
Exercises
Edexcel GCSE Mathematics Textbook
Page 111 – Exercise 8D
Q1 (right column), Q2 (right column)
Expanding two brackets
1
2
3
4
5
6
7
8
9
10
11
12
13
14
𝑥 + 1 𝑥 + 2 = 𝑥 2 + 3𝑥?+ 2
?2
𝑥 + 2 𝑥 − 1 = 𝑥2 + 𝑥 −
𝑥 − 3 𝑥 − 4 = 𝑥 2 − 7𝑥?+ 12
𝑥 + 1 2 = 𝑥 2 + 2𝑥?+ 1
𝑥 − 5 2 = 𝑥 2 − 10𝑥
? + 25
𝑥 − 10 2 = 𝑥 2 − 20𝑥
? + 100
𝑥 + 𝑦 𝑥 + 2𝑦 = 𝑥 2 + 3𝑥𝑦? + 2𝑦 2
? − 𝑞2
𝑝 + 𝑞 3𝑝 − 𝑞 = 3𝑝2 + 2𝑝𝑞
? + 2𝑦 2
3𝑥 − 2𝑦 𝑥 − 𝑦 = 3𝑥 2 − 5𝑥𝑦
𝑎 + 𝑏 𝑐 + 𝑑 = 𝑎𝑐 + 𝑎𝑑 +
? 𝑏𝑐 + 𝑏𝑑
2𝑥 + 3𝑦 3𝑥 − 4𝑦 = 6𝑥 2 + 𝑥𝑦?− 12𝑦 2
𝑎 − 𝑏 𝑎 + 𝑏 = 𝑎2 − 𝑏 2 ?
𝑥 + 𝑦 2 = 𝑥 2 + 2𝑥𝑦
? + 𝑦2
2𝑥 − 3𝑦 2 = 4𝑥 2 − 12𝑥𝑦
? + 9𝑦 2
Faster expansion of squared brackets
There’s a quick way to expand squared brackets involving two terms:
𝑥+𝑦
2
= 𝑥 2 + 2𝑥𝑦
? + 𝑦2
2𝑥 − 𝑦
2
? + 𝑦2
= 4𝑥 2 − 4𝑥𝑦
3𝑎𝑏 + 4𝑐
2
= 9𝑎2 𝑏 2 + 24𝑎𝑏𝑐
? + 16𝑐 2
7𝑥𝑦 − 2𝑧
2
= 49𝑥 2 𝑦 2 − 28𝑥𝑦𝑧
+ 4𝑧 2
?
Four different types of factorisation
1. Factoring out a term
2𝑥 2 + 4𝑥 = 2𝑥 𝑥 +? 2
2. 𝒙𝟐 + 𝒃𝒙 + 𝒄
𝑥 2 + 4𝑥 − 5 = 𝑥 + 5 ? 𝑥 − 1
3. Difference of two squares
4. 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄
4𝑥 2 − 1 = 2𝑥 + 1 ?2𝑥 − 1
? − 1)
2𝑥 2 + 𝑥 − 3 = (2𝑥 + 3)(𝑥
Strategy: either split the middle
term, or ‘go commando’.
2. 𝑥 2 + 𝑏𝑥 + 𝑐
Which is (𝒙 + 𝒂)(𝒙 + 𝒃)?
How does this suggest we can factorise say 𝑥 2 + 3𝑥 + 2?
𝑥 2 − 𝑥 − 30 = 𝑥 + 5 ? 𝑥 − 6
Is there a good strategy for working out which
numbers to use?
2. 𝑥 2 + 𝑏𝑥 + 𝑐
1
2
3
𝝅
4
5
6
7
8
9
10
𝑥 2 + 4𝑥 + 3 = 𝑥 + 3 ? 𝑥 + 1
𝑥 2 − 8𝑥 + 7 = 𝑥 − 1 ? 𝑥 − 7
𝑥 2 + 2𝑥 − 8 = 𝑥 + 4 ? 𝑥 − 2
𝑥 2 + 16𝑥 − 36 = 𝑥 + 18? 𝑥 − 2
𝑦 2 − 𝑦 − 56 = (𝑦 + 7)(𝑦
? − 8)
𝑧 2 + 3𝑧 − 54 = 𝑧 + 9 ?𝑧 − 6
𝑧 2 − 3𝑧 − 54 = 𝑧 − 6 ?𝑧 + 9
𝑧 2 + 15𝑧 + 54 = 𝑧 + 6 ?𝑧 + 9
𝑥 2 + 4𝑥 + 4 = 𝑥 + 2 2?
𝑥 2 − 14𝑥 + 49 = 𝑥 − 7 2?
𝑥 4 + 5𝑥 2 + 4 = 𝑥 2 + 1 ?𝑥 2 + 4
3. Difference of two squares
Firstly, what is the square root of:
4𝑥 2 = 2𝑥 ?
25𝑦 2 = 5𝑦 ?
16𝑥 2 𝑦 2 = 4𝑥𝑦?
𝑥 4 𝑦 4 = 𝑥 2 𝑦?2
9 𝑧−6
2
? 6)
= 3(𝑧 −
3. Difference of two squares
3
3
2𝑥
2𝑥
2
4𝑥 − 9
=(
+
)(
Click to Start
Bromanimation
−
)
3. Difference of two squares
2
? − 𝑥)
1 − 𝑥 = (1 + 𝑥)(1
𝑥+1
2
49 − 1 − 𝑥
(In your head!)
− 𝑥−1
2
2
= 4𝑥
?
? + 𝑥)
= (8 − 𝑥)(6
512 − 492 = 200?
18𝑥 2 − 50𝑦 2 = 2 3𝑥 + 5𝑦 ? 3𝑥 − 5𝑦
2𝑡 + 1
2
−9 𝑡−6
2
= 5𝑡 − 17 ?−𝑡 + 19
3. Difference of two squares
Exercises:
1
2
3
4
5
6
7
8
9
10
4𝑝2 − 1 = 2𝑝 + 1 ? 2𝑝 − 1
4 − 𝑥 2 = (2 + 𝑥)(2
? − 𝑥)
144 − 𝑏 2 = 12 + 𝑏 ? 12 − 𝑏
𝑥 + 1 2 − 25 = 𝑥 + 6 ?𝑥 − 4
?
7.642 − 2.362 = 52.8
2𝑝2 − 32 = 2 𝑝 + 4 ? 𝑝 − 4
3𝑦 2 − 75𝑥 2 = 3 𝑦 + 5𝑥? 𝑦 − 5𝑥
4𝑎2 − 64𝑏 2 = 4 𝑎 + 4𝑏 ? 𝑎 − 4𝑏
9 𝑝 + 1 2 − 4𝑝2 = 5𝑝 + 3 ?𝑝 + 3
50 2𝑥 + 1 2 − 18 1 − 𝑥 2 = 2(7𝑥 + 8)(13𝑥
+ 2)
?
4. 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
2
2𝑥
?
+ 𝑥 − 3 = (2𝑥 + 3)(𝑥
− 1)
Factorise using:
a. The ‘commando’ method*
b. Splitting the middle term
* Not official mathematical terminology.
4. 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
2𝑥 2 + 11𝑥 + 12 = 𝑥 + 4 2𝑥 ?+ 3
6𝑥 2 − 7𝑥 − 3 = (2𝑥 − 3)(3𝑥? + 1)
? 3𝑦
2𝑥 2 − 5𝑥𝑦 + 3𝑦 2 = 𝑥 − 𝑦 2𝑥 −
6𝑥 2 − 3𝑥 − 3 = 3(𝑥 − 1)(2𝑥? + 1)
Exercises
1
2
3
4
5
6
7
8
9
10
11
N
N
? + 1)
2𝑥2 + 3𝑥 + 1 = (2𝑥 + 1)(𝑥
3𝑥2 + 8𝑥 + 4 = (3𝑥 + 2)(𝑥
? + 2)
2𝑥2 − 3𝑥 − 9 = (2𝑥 + 3)(𝑥
? − 3)
4𝑥2 − 9𝑥 + 2 = (4𝑥 − 1)(𝑥
? − 2)
2𝑥2 + 𝑥 − 15 = (2𝑥 − 5)(𝑥
? + 3)
2𝑥2 − 3𝑥 − 2 = (2𝑥 + 1)(𝑥
? − 2)
3𝑥2 + 4𝑥 − 4 = (3𝑥 − 2)(𝑥
? + 2)
6𝑥 2 − 13𝑥 + 6 = 3𝑥 − 2 ? 2𝑥 − 3
15𝑦 2 − 13𝑦 − 20 = 5𝑦 + 4 ? 3𝑦 − 5
12𝑥 2 − 𝑥 − 1 = 4𝑥 + 1 ?3𝑥 − 1
25𝑦 2 − 20𝑦 + 4 = 5𝑦 − ?2 2
Well Hardcore:
4𝑥 3 + 12𝑥 2 + 9𝑥 = 𝑥 2𝑥 +?3
𝑎2 𝑥 2 − 2𝑎𝑏𝑥 + 𝑏 2 = 𝑎𝑥 −?𝑏
2
2
‘Commando’ starts
to become difficult
from this question
onwards.
Simplifying Algebraic Fractions
2𝑥 2 + 4𝑥
2𝑥
?
=
𝑥2 − 4
𝑥−2
3𝑥 + 3
3
= ?
2
𝑥 + 3𝑥 + 2 𝑥 + 2
2𝑥 2 − 5𝑥 − 3
2𝑥 + 1
=− ? 3
3
4
6𝑥 − 2𝑥
2𝑥
Negating a difference
− 4 − 𝑦 = 𝑦 −? 4
− 2𝑥 − 9 = 9 −? 2𝑥
1−𝑥
= −1 ?
𝑥−1
3 − 2𝑥 2 − 𝑥
𝑥−2
?
=
2𝑥 − 3 𝑥 + 1
𝑥+1
Exercises
1
2
3
4
5
6
2𝑥 + 6 𝑥 + 3
= ?
2𝑥
𝑥
7
2𝑥 2 − 8
2 𝑥−2
?
=
2
𝑥 + 6𝑥 + 8
𝑥+4
4𝑥 + 8 4
= ?
3𝑥 + 6 3
8
𝑥 2 + 2𝑥 𝑥
= ?
8𝑥 + 16 8
𝑥 2 + 5𝑥 + 6 𝑥 + 2
= ?
2
𝑥 +𝑥−6
𝑥−2
9
𝑥2 − 9
𝑥+3
= ?
2
2𝑥 − 7𝑥 + 3 2𝑥 − 1
2𝑥 + 10
2
= ?
2
𝑥 − 25 𝑥 − 5
𝑥+3
1
= ?
2
𝑥 −9 𝑥−3
𝑥2 + 𝑥 − 2 𝑥 − 1
= ?
2
𝑥 −4
𝑥−2
10
11
6𝑥 2 − 𝑥 − 1 3𝑥 + 1
= ?
2
4𝑥 − 1
2𝑥 + 1
2𝑦 2 + 4𝑦
9𝑦 2 − 1
× 2
=2 ?
2
3𝑦 + 7𝑦 + 2 3𝑦 − 𝑦
Algebraic Fractions
3 1
7
+
= ?
5 10 10
2 1
5
− = ?
3 4 12
How did we identify the new denominator to use?
(Note: If you’ve added/subtracted fractions before using
some ‘cross-multiplication’-esque method, unlearn it now,
because it’s pants!)
Algebraic Fractions
The same principle can be applied to algebraic fractions.
1 2
+ 2=
𝑥 𝑥
𝑥
2
+ 2
2
𝑥
𝑥
?
𝑥+2
= 2
𝑥
1
2
1
− 2
=
?
𝑥 𝑥 + 2𝑥
𝑥+2
1
1
1
?
− =−
𝑥+1 𝑥
𝑥+1
1
1
2
2
?
+
−
=
3𝑥 + 6 5𝑥 + 10 15𝑥 + 30 5 𝑥 + 2
5
3
4 𝑥+3
−
=
?
2𝑥 + 1 2𝑥 + 3
2𝑥 + 1 2𝑥 + 3
“To learn the secret ways of
algebra ninja, simplify
fraction you must.”
Recap
1
1
3
+
=
?1
2𝑥 + 2 𝑥 + 1 2 𝑥 +
1
1 1 + 𝑥𝑦
+ =
𝑥𝑦 2 𝑦
𝑦?
1 𝑥+1
1+ =
?
𝑥
𝑥
𝑥
3
2𝑥 2 + 𝑥 − 3
?
−
=
𝑥 + 1 2𝑥 + 1
𝑥 + 1 2𝑥 + 1
1
1
1+𝑥
+
=
?
2
𝑥 +𝑥 𝑥+1 𝑥 𝑥+1
Exercises
1
2
3
4
5
3
1
1
−
=
5 𝑥+1
2 𝑥+1
10 𝑥 +?1
1
2
11
+
=
4𝑥 3𝑥 12𝑥
?
2
4
2𝑥 − 2
−
=
?
𝑥 − 1 𝑥2 − 1 𝑥2 − 1
8
1 2𝑥 + 1
2+ =
?
𝑥
𝑥
9
1
𝑥4 + 1
𝑥 + 2=
?
𝑥
𝑥2
2
1
𝑥
=
𝑥+1 𝑥+1
2
1
𝑥−1
+
=
𝑥2 − 9 𝑥 + 3
𝑥 + 3 ?𝑥 − 3
10
1−
2
4
2𝑥
−
=
2−𝑥 4−𝑥
2 − 𝑥 ?4 − 𝑥
11
2
1
+
4𝑥 2 − 4𝑥 − 3 4𝑥 2 + 8𝑥 + 3
6
3
4
+
𝑥+1
𝑥+1
7
1
2
𝑥+5
?
−
=
𝑥 − 3 3𝑥 − 1
𝑥 − 3 3𝑥 − 1
2
3𝑥 + 7
=
?
𝑥+1 2
?
?
Completing the Square – Starter
Expand the following:
𝑥+3
2
= 𝑥 2 + 6𝑥?+ 9
𝑥+5
2
? + 26
+ 1 = 𝑥 2 + 10𝑥
𝑥−3
2
= 𝑥 2 − 6𝑥?+ 9
𝑥+𝑎
2
? + 𝑎2
= 𝑥 2 + 2𝑎𝑥
What do you notice about the coefficient of the 𝑥
term in each case?
Completing the square
Typical GCSE question:
“Express 𝑥 2 + 6𝑥 in the form 𝑥 + 𝑝
and 𝑞 are constants.”
𝑥+3
?
2
2
+ 𝑞, where 𝑝
−9
Completing the square
More examples:
2
2
?
𝑥 − 2𝑥 = 𝑥 − 1 − 1
𝑥 2 − 6𝑥 + 4 = 𝑥 − 3 ?2 − 5
𝑥 2 + 8𝑥 + 1 = 𝑥 + 4 2? − 15
𝑥 2 + 10𝑥 − 3 = 𝑥 + 5 2? − 28
2
2
𝑥 + 4𝑥 + 3 = 𝑥 + 2 ? − 1
2
2
?
𝑥 − 20𝑥 + 150 = 𝑥 − 10 + 50
Exercises
Express the following in the form 𝑥 + 𝑝
1
2
3
4
5
6
7
8
2
+𝑞
𝑥 2 + 2𝑥 = 𝑥 + 1 ?2 − 1
𝑥 2 + 12𝑥 = 𝑥 + 6 ?2 − 36
𝑥 2 − 22𝑥 = 𝑥 − 11 ?2 − 121
𝑥 2 + 6𝑥 + 10 = 𝑥 + 3 2? + 1
𝑥 2 + 14𝑥 + 10 = 𝑥 + 7 2? − 39
𝑥 2 − 2𝑥 + 16 = 𝑥 − 1 2?+ 15
𝑥 2 − 40𝑥 + 20 = 𝑥 − 20 ?2 − 380
2
1
1
11
2
?
𝑥 +𝑥 = 𝑥+
−
2
4
2
9
5
29
2
𝑥 + 5𝑥 − 1 = 𝑥 + ? −
2
4
2
10
𝑥2
9
1
− 9𝑥 + 20 = 𝑥 − ? −
2
4
𝑥 2 + 2𝑎𝑥 + 1 = 𝑥 + 𝑎
2
?− 𝑎2 + 1
More complicated cases
Express the following in the form 𝑎 𝑥 + 𝑝
2
+ 𝑞:
3𝑥 2 + 6𝑥 = 3 𝑥 + 1?2 − 3
2𝑥 2 + 8𝑥 + 10 = 2 𝑥 + 2 ?2 + 2
? 2+6
−𝑥 2 + 6𝑥 − 3 = −1 𝑥 − 3
5𝑥 2 − 30𝑥 + 5 = 5 𝑥 − 3 ?2 − 40
−3𝑥 2 + 12𝑥 − 6 = −3 𝑥 − ?2 2 + 6
1 − 24𝑥 − 4𝑥 2 = −4 𝑥 − ?3 2 + 37
Exercises
Put in the form 𝑎 𝑥 + 𝑝
1
2
3
4
5
6
7
2
+ 𝑞 or 𝑞 − 𝑎 𝑥 + 𝑝
2𝑥 2 + 4𝑥 = 2 𝑥 + 1 2?− 2
2𝑥 2 − 12𝑥 + 28 = 2 𝑥 − 3 2?+ 10
3𝑥 2 + 24𝑥 − 10 = 3 𝑥 + 4 2?− 58
5𝑥 2 + 20𝑥 − 19 = 5 𝑥 + 2 2?− 39
−𝑥 2 + 2𝑥 + 16 = 17 − 𝑥 −?1
9 + 4𝑥 − 𝑥 2 = 13 − 𝑥 −?2 2
2
13
3
2
1 − 3𝑥 − 𝑥 =
− 𝑥+
?2
4
2
2
Proofs
Show that for any integer 𝑛,
2
𝑛 + 𝑛 is always even.
How many 𝑛 would we need to try before
we’re convinced this is true? Is this a good
approach?
Proofs
Prove that the sum of three consecutive
integers is a multiple of 3.
We need to ensure this works for any
possible 3 consecutive numbers. What could
we represent the first number as to keep
things generic?
Proofs
Prove that odd square numbers are always 1
more than a multiple of 4.
How would you represent…
Any odd number:
2𝑛 ?+ 1
Any even number:
?
2𝑛
Two consecutive
odd numbers.
2𝑛 + 1,?2𝑛 + 3
Two consecutive
even numbers.
? +2
2𝑛 , 2𝑛
One less than a
multiple of 3.
3𝑛 ?− 1
Proofs
Prove that the difference between the squares
of two odd numbers is a multiple of 8.
Example Problems
People in the left
row work on this:
[June 2012] Prove that
2𝑛 + 3 2 − 2𝑛 − 3 2 is a
multiple of 8 for all positive
integer values of 𝑛.
People in the middle
row work on this:
People in in the right
row work on this:
[Nov 2012] (In the previous
part of the question, you
were asked to factorise
2𝑡 2 + 5𝑡 + 2, which is (2𝑡 +
1)(𝑡 + 2) )
[March 2013] Prove
algebraically that the
difference between the
squares of any two
consecutive integers is
equal to the sum of these
two integers.
“𝑡 is a positive whole
number. The expression
2𝑡 2 + 5𝑡 + 2 can never be a
prime number. Explain why.”
Exercises
Edexcel GCSE Mathematics Textbook
Page 469 – Exercise 28E
Odd numbered questions
Even/Odd Proofs
Some proofs don’t need algebraic manipulation. They just require us to reason
about when our number is odd and when our number is even.
Prove that 𝑛2 + 𝑛 + 1 is always odd for all integers 𝑛.
When 𝑛 is even:
𝑛2 is 𝑒𝑣𝑒𝑛 × 𝑒𝑣𝑒𝑛 = 𝑒𝑣𝑒𝑛. So 𝑛2 + 𝑛 + 1 is 𝑒𝑣𝑒𝑛 +
𝑒𝑣𝑒𝑛 + 𝑜𝑑𝑑 = 𝑜𝑑𝑑.
When 𝑛 is odd:
?
𝑛2 is 𝑜𝑑𝑑 × 𝑜𝑑𝑑 = 𝑜𝑑𝑑. So 𝑛2 + 𝑛 + 1 is 𝑜𝑑𝑑 + 𝑜𝑑𝑑 +
𝑜𝑑𝑑 = 𝑜𝑑𝑑.
Therefore 𝑛 is always odd.