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Algebra by Mind Map: Algebra

1. Factorisation

1.1. Always take out Highest Common Factor first

1.2. For 2 terms,

1.2.1. As per expression, no change

1.2.2. Difference Of Two Squares

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

1.3. For 3 terms,

1.3.1. Perfect squares

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

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

1.3.2. Cross Multiplication/multiplication frame

1.4. For 4 terms

1.4.1. Grouping

1.4.2. NOTE: (b-a) can be rewritten as -(a-b)

2. Expansion

2.1. FOIL distributive property

2.1.1. Always remember to multiply by sign in front of term.

2.1.2. FOIL is the acronym for First Outer Inner Last

2.1.3. Work the brackets from the first numbers, followed by outer, inner and last

2.1.4. Example: (a+b)(c+d) = ac +ad +bc +bd

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)2 = a2 - 2ab + b2

2.2.1.3. Common mistakes: (a+b)2 = a2+b2 or (a-b)2= a2-b2

2.2.2. Difference of 2 Squares

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

3. Applications to real life

3.1. Calculating grocery expenses You need algebra to estimate the amount of money needed to pay for your grocery so that you’ll know if someone overcharges you or if you don’t have enough money.

3.1.1. For example, if you buy milk and cheese (a+b) and you have a 10%+20% discount (10/100+20/100) you can use the FOIL method.

3.1.2. using algebra to figure out how many items you can buy eg. if you have $30 and want to find out how many bottles of soda ($2 each) you can buy, you can use an algebraic equation 2x=30 to figure it out.

3.2. To save money and find cheaper deals

3.3. Calculating area of house property

3.3.1. For example, if you're buying furniture for your new house, you will need an estimate of how big the living room, dinning room, bed room etc in order to buy the proper furniture to suit your house. We can use expansion and factorisation to make this easier for us

3.3.2. If you are an architect, you have to calculate the possible lengths and widths of the building you are going to build according to the land size

3.4. Filling up the gas tank You need algebra to know how much gas you can buy. For example, 1 gallon of gas cost $2 and you only have $10 so you can only buy 5 gallons of gas.

4. Solutions to quadratic equations

4.1. Zero Product Principle

4.1.1. step 1: ensure expression = 0

4.1.2. step 2: factorise LHS

4.1.3. step 3: apply Zero Product Principle

4.1.4. step 4: solve the equation

4.1.5. must be in the form (factor)(factor)

4.2. 3 terms without any common factor

4.2.1. x2-4x-5=0 (x-5)(x+1)=0 x-5 = 0 or (x+1) = 0

4.3. Terms with common factor

4.3.1. x2-5x=0 x(x-5)=0 x=0 or x-5=0

4.4. 2 terms which involve difference of two squares

4.4.1. x2-49=0 (x=7)(x-7)=0 x-7=0 x=7 or x+7= 0 x=-7

4.5. 2 terms which involve taking square root on both sides

4.5.1. x2-25 =0 x2=25 x=+- square root 5 x=+-5