# Algebra

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Algebra

## 1. Factorisation

### 1.1. 2 Terms

1.1.1. Difference of 2 Squares

1.1.1.1. (a)2 - (b)2 = (a+b)(a-b)

1.1.1.2. HCF

### 1.2. 3 Terms

1.2.1. Perfect Squares

1.2.1.1. a2 + 2ab + b2 = (a+b)2

1.2.1.2. a2 + 2ab + b2 = (a-b)2

1.2.2. Quadratic expressions

1.2.2.1. Cross multiplication

1.2.2.2. multiplication frame

### 1.3. 4 Terms

1.3.1. Grouping Method

## 2. Expansion

### 2.1. FOIL

2.1.1. First Outer Inner Last

2.1.1.1. (a+ b)(c+d) = ac + ad + bd + bc

### 2.2. Algebraic Identities

2.2.1. Perfect Square

2.2.1.1. (a+b)2 = a2 + 2ab + b2

2.2.1.2. (a-b) = a2 - 2ab + b2

2.2.2. Difference of 2 Squares

2.2.2.1. (a-b)(a+b) = a2 - b2

## 3. Solving Quadratic Equations

### 3.1. 3 terms without any common factor

3.1.1. x2 - 4x -5 = 0 (x-5)(x-1) = 0

### 3.2. terms with a common factor

3.2.1. x2 - 5x = 0 x(x-5) = 0

### 3.3. 2 terms that involve difference of 2 squares

3.3.1. x2 -49 = 0 (x+7)(x-7) = 0

### 3.4. 2 terms which involve taking square root on both sides

3.4.1. answer can be negative or positive

### 3.5. Steps to solve

3.5.1. 1. Ensure expression is 0 on the left side

3.5.2. 2. Factorise the left side

3.5.3. 5. Apply zero product principle

3.5.4. 4. Solve equation

### 3.6. Zero product principle

3.6.1. a x b = 0, so either a = 0, b = 0 or a = b = 0

## 4. Real life Applications

### 4.1. Factorising

4.1.1. Computer security

4.1.1.1. Algorithms used to ensure “secure” digital communications is based on the difficulty of factorizing a large number when all its prime components (excluding “1”) are unknown, and on how easy it is to find a second factor given a first factor.

4.1.2. Shopping

4.1.2.1. I bought 5 bags of sweets and there are 2 apple- flavoured and 3 pear- flavoured sweets in each bag. Using factorisation, i can calculate the number each flavour of sweets I have by expressing it as 5(2a + 3p).

### 4.2. Expansion

4.2.1. gasoline effeciency

4.2.1.1. Practical Applications of Mathematics in Everyday Life

4.2.1.1.1. (6.5lbs)(30% water) + (Xlbs)(100% water) = (6.5lbs + Xlbs)(50% Water)