ALGEBRA
by NATANIA LEOW EN-QI 1G
## 1. expansion

### 1.1. algebraic identities

1.1.1. perfect square

1.1.1.1. (a+b)^2 =a^2+2ab+b^2

1.1.1.2. (a-b)^2 =a^2-2ab+b^2

1.1.1.3. (a-b)(a+b) = a^2 - b^2

1.1.2. difference of two squares

1.1.2.1. a^2 - b^2

### 1.2. FOIL

1.2.1. stands for First Outer Inner Last

### 1.3. 1. Simplify brackets first 2. Expand again (e.g one bracket within another)

## 2. factorisation

### 2.1. 1. Take out HCF

2.1.1. 2a) 2 terms

2.1.1.1. Difference of 2 squares

2.1.1.1.1. a^2-b^2 = (a-b)(a+b)

2.1.2. 2b) 3 terms

2.1.2.1. Algebraic Identity

2.1.2.1.1. a^2+2ab+b^2 = (a+b)^2

2.1.2.2. Multiplication frame or Cross-Multiplication method

2.1.3. 2c) 4 terms

2.1.3.1. Grouping Method

## 3. Things to note

### 3.1. a-b= -(b-a)

### 3.2. -(a-b)=(b-a)

### 3.3. Don't cancel indices

### 3.4. Multiply by sign IN FRONT of term(remember especially for FOIL method)

### 3.5. 1 = 1^2

## 4. Solutions for 4 types of common quadratic equations

### 4.1. 3 terms without any common factor

4.1.1. 1. ensure expression on the left hand side = 0

4.1.1.1. 1. x^2-4x-5=0 2. (x-5)(x+1)=0 3. x-5=0 or x+1=0 4. x=5 or x=-1

4.1.2. 2. factorise left hand side

4.1.3. 3. Apply "Zero Product Principle"

4.1.4. 4. Solve the equation

### 4.2. terms with a common factor

4.2.1. 1. x^2-5x=0 2. x(x-5)=0 3. x=0 or x-5=0 4. x=0 or x=5

### 4.3. 2 terms which involve difference of 2 squares

4.3.1. 1. x^2-49=0 2. (x+7)(x-7)=0 3. x+7=0 or x-7+0 4. x=-7 or x=7

### 4.4. 2 terms which involve taking square root on both sides

4.4.1. 1. x^2-25=0 2. x^2=25 3. x=±√25 x=±5

### 4.5. *Don't divide both sides by a variable because you'll lose the value of the variable

### 4.6. *To use zero product principle, RHS MUST be 0.

## 5. Real Life Applications

### 5.1. I want to sell a product, mixed nuts.I try to find the ideal size I want for the box or can for the quantity of product I want to sell. After all, cardboard and metal costs money, and storage of overly large containers wastes cash. To find the ideal size, I will need to be able to factorise. Also, those nuts I want to mix; prices on nuts change all the time. One day the cost of peanuts may be up, or on another day walnuts may be down. How should I tweak the mix to hold the price I charge constant as the various nut prices change? To figure that out, I will need to know about factoring.

### 5.2. Well, suppose you would like to own a business one day. Say you own a painting company and have several employees. You get a rush job to paint a large hotel conference room. Knowing from experience how fast your employees work, you know that Joe can do a room this size in twelve hours, Max can do the job in nine, and Jane can do the job in ten and a half. How long should it take them, then, to do the whole job if you let them work together? To figure this out you need to be able to factor.

### 5.3. expanding two brackets is a skill needed for graphing and analysing 'parabola' shapes such as Sydney Harbor Bridge as seen below!!

### 5.4. <img src="http://passyworldofmathematics.com/Images/pwmImagesFive/BinomialThree550x484JPG.jpg" alt="Real World Expanding Two Bracket Binomials 3"/>