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. Solutions for 4 types of common quadratic equations
2.1. 3 terms without any common factor
2.1.1. 1. ensure expression on the left hand side = 0
2.1.1.1. 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
2.1.2. 2. factorise left hand side
2.1.3. 3. Apply "Zero Product Principle"
2.1.4. 4. Solve the equation
2.2. terms with a common factor
2.2.1. 1. x^2-5x=0 2. x(x-5)=0 3. x=0 or x-5=0 4. x=0 or x=5
2.3. 2 terms which involve difference of 2 squares
2.3.1. 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
2.4. 2 terms which involve taking square root on both sides
2.4.1. 1. x^2-25=0 2. x^2=25 3. x=±√25 x=±5
2.5. *Don't divide both sides by a variable because you'll lose the value of the variable
2.6. *To use zero product principle, RHS MUST be 0.
3. Real Life Applications
3.1. 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.
3.2. 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.
3.3. expanding two brackets is a skill needed for graphing and analysing 'parabola' shapes such as Sydney Harbor Bridge as seen below!!
3.4. <img src="http://passyworldofmathematics.com/Images/pwmImagesFive/BinomialThree550x484JPG.jpg" alt="Real World Expanding Two Bracket Binomials 3"/>
4. factorisation
4.1. 1. Take out HCF
4.1.1. 2a) 2 terms
4.1.1.1. Difference of 2 squares
4.1.1.1.1. a^2-b^2 = (a-b)(a+b)
4.1.2. 2b) 3 terms
4.1.2.1. Algebraic Identity
4.1.2.1.1. a^2+2ab+b^2 = (a+b)^2
4.1.2.2. Multiplication frame or Cross-Multiplication method
4.1.3. 2c) 4 terms
4.1.3.1. Grouping Method