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Long Multiplication

Traditional Method:                                           Alternative Method:                                        

158 x 67                                                   158 x 67 

158                                                          x   100      50     8

x67                                                         60   60000  3000  480 

-----                                                        7    700      350   56

1106 (158 x 7)                                         6000 + 3000 + 480 + 700  350 + 56 = 

9470 (158 x 60)                                       11586

-----

10586 (1106 + 9470) 

(Also known as column method)                 (Also known as box method) 

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Combined (Conditional) Probability

Rachel has a bag of 5 red and 4 green marbles. She picks one out of the bag and records the colour. Then she picks another one out of the bag and also records the colour. What is the probability that both the marbles are red? 

First marble is red: 5/9                  Second marble is red: 5/8 (1st b) or 4/8 (2nd b)

First marble is green: 4/9               Second marble is green: 3/8 (1st b) or 4/8 (2nd                                                   b)

Both marbles are red: 

5/9 x 5/8 = 25/72

One marble of each colour:

(5/9 x 3/8) + (4/9 x 4/8) = 15/72 + 16/72 = 31/72

Both marbles are green:

4/9 x 3/8 = 12/72 

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Quadratic Equations

Show that x squared (sqd) - 4x - 12 = 0

x sqd + (x + 2) sqd = (x + 4) sqd

x sqd + (x + 2) ( x + 2) = (x + 4) (x + 4)                   times out the brackets 

x sqd + x sqd + 4x (2x + 2x) + 4 x sqd +  8x (4x + 4x) + 16

x sqd + 4x + 4 = 8x + 16                                          take away x sqd

x sqd - 4x + 4 = 16                                                  take away 8x

x sqd - 4x - 12 = 0                                                   take away 16

Solve x sqd - 4x - 12 = 0

(x - 2) sqd - 16 = 0   ~ (x - 2) sqd = 16 ~ x - 2 = sqd root of 16 ~ x = sqd root of 16 + 2 or sqd root of 16 - 2  

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