Class notes Mathematics

Formula List
SURDS, INDICES, LOGRITHM
Surds

1 𝑥 𝑦 𝑧 = 𝑥𝑦√𝑧, e.g., √180 = √3 × 2 × 5 = 3 × 2 × √5

2 𝒙 = √𝑥 , 𝒙 = √𝑥

√
3 √𝑎 × 𝑏 = √𝑎 × √𝑏 4 =
√
√ √
5 = × = 6 = ×
√ √
=
√ √
√ √ √ √ √ √ √ √ √

7 √𝑎 + 𝑏 ≠ √𝑎 + √𝑏 8 √𝑎 − 𝑏 ≠ √𝑎 − √𝑏

Indices

1 𝑎 ×𝑎 =𝑎 2 =𝑎

3 𝑎 ×𝑏 = (𝑎 × 𝑏) 4 =

5 (𝑎 ) = 𝑎 6 𝑎 =1

7 𝑎 = 8 =

9 𝑎 = √𝑎 10 √𝑎 = 𝑎 = √𝑎

11 (𝑎 + 𝑏) ≠ 𝑎 + 𝑏 , e.g., (𝑎 + 𝑏) = 𝑎 + 2𝑎𝑏 + 𝑏

𝑎 𝑏
12 =
𝑏 𝑎
Logarithm

1 Logarithmic form: log 𝑁 = 𝑝 𝑁 > 0, 𝑎 > 0, 𝑎 ≠ 1. Index form: 𝑎 = 𝑁.

2 log 𝑥 + log 𝑦 = log 𝑥𝑦 3 log 𝑥 − log 𝑦 = log

4 log 𝑥 = 𝑘 log 𝑥 5 log = − log 𝑥

6 log 𝑏 = 7 log 𝑏 =

8 log 𝑎 = 1 9 log 1 = 0

10 lg 𝑥 = log 𝑥 11 ln 𝑥 = log 𝑥

12 Single logarithm: 2 log 𝑀 − 3 log 𝑁 + 4 log 𝑃 = log

13 𝑒 =𝑥 10 =𝑥 𝑎 =𝑥




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, Formula List
ALGEBRA
1. (a) Remainder Theorem (b) Factor Theorem
For 𝑓(𝑥) = … … divided by (𝑥 + 𝑎), For 𝑓(𝑥) = … … divided by (𝑥 + 𝑎),
𝑅 = 𝑓(−𝑎). 𝑅 = 𝑓(−𝑎).
If 𝑅 = 0, (𝑥 + 𝑎) is a factor.
2. 𝒙𝟐 − 𝟓𝒙 + 𝟒 is a factor of 𝒇(𝒙),
then (𝑥 − 1) is a factor of 𝑓(𝑥)
and (𝑥 − 4) is a factor of 𝑓(𝑥).
3. To factorize a cubic expression 𝒇(𝒙) = 𝒂𝒙𝟑 + 𝒃𝒙𝟐 + 𝒄𝒙 + 𝒅
Step 1: Try 𝑓(2) = 0, (𝑥 − 2) is a factor.
Step 2: Divide 𝑓(𝑥) by (𝑥 − 2).
Step 3: 𝑓(𝑥) = (𝑥 − 2)(𝑎𝑥 + 𝑏𝑥 + 𝑐)
= (𝑥 − 2)(𝑥 + 𝑝)(𝑥 + 𝑞)
4. Shape of curve:

Linear Quadratic Cubic




+𝑥 −𝑥 +𝑥 −𝑥
5. Forming equation:

9


−1 5
−1 1 3

−10


Roots: −1 and 5, Roots: −1, 1 and 3,
𝑓(𝑥) = 𝑘(𝑥 + 1)(𝑥 − 5) 𝑓(𝑥) = 𝑘(𝑥 + 1)(𝑥 − 1)(𝑥 − 3)
→ 𝑘. 1(−5) = −10 → 𝑘. 1. (−1)(3) = 9
→ 𝑘 = −2 →𝑘=3
𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛: ∴ 𝑓(𝑥) = 2(𝑥 + 1)(𝑥 − 5) 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛: ∴ 𝑓(𝑥) = 3(𝑥 + 1)(𝑥 − 1)(𝑥 − 3)

6. (i) 𝑓(𝑥) = (𝑥 − 𝑎)(𝑥 − 𝑏) (ii) 𝑔(𝑥) = (𝑥 − 𝑎)(𝑥 − 𝑏)(𝑥 − 𝑐)
= 𝑥 + … … … … + (−𝑎)(−𝑏) = 𝑥 + … … … … + (−𝑎𝑏𝑐)




4

, Formula List
QUADRATIC EQUATION
❶ Completing Square
Minimum curve Maximum curve
𝑓(𝑥) = 𝐴(𝑥 + 𝐵) + 𝐶 𝑓(𝑥) = −𝐴(𝑥 + 𝐵) + 𝐶
 Minimum value of 𝑓(𝑥) = 𝐶  Maximum value of 𝑓(𝑥) = 𝐶
when 𝑥 = −𝐵 when 𝑥 = −𝐵
 Maximum value of ( )
=  Minimum value of ( )
=
when 𝑥 = −𝐵 when 𝑥 = −𝐵

❷
For 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 > 𝟎 For 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 < 𝟎
Critical values = −2, 5 Critical values = −2, 5


+ +

−2 5 −2 5
− −
𝑥 < −2 𝑥>5
−2 < 𝑥 < 5
Less than Less Greater than Greater
❸

Nature of Roots



Real Unreal

𝑏 − 4𝑎𝑐 ≥ 0 𝑏 − 4𝑎𝑐 < 0




Real and Different Real and Equal

𝑏 − 4𝑎𝑐 > 0 𝑏 − 4𝑎𝑐 = 0


❹
Line and Curve

𝑦 = 𝑓(𝑥) ------------- (1)
𝑦 = 𝑔(𝑥) ------------- (2)
Step – 1: Form Quadratic Equation
Step – 2: Use Discriminant





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