Transition Year Maths Sociable Numbers:


These are groups of three or more numbers which form closed loops. The sum of the divisors of the first give the second, the sum of the divisors of the second give the third and so on until the divisors of the last give the first number.


PROJECT 16.9 Loop of Five


12,496 is the first in a loop of five sociable numbers. Working in small groups of 2 - 4 try to find the other four numbers in this loop. This could take 1 hour or more. (See hint on page 155.)


Factorising Quadratics using:


x bb ac a


= − ± 2 − 4 2 Example 1: Multiply (x + 4) by (x + 5)


Each part of the first bracket must be multiplied by each part of the second bracket. ⇒ x by x = x2 x by 5 = 5x 4 by x = 4x 4 by 5 = 20


∴ Multiply (x + 4)(x + 5) = x2 + 9x + 20


+ 9x + 20 are (x + 4) and (x + 5). These may not be the only factors. It is easy to multiply two expressions like (x + 4) and (x + 5) to get x2


+ 9x + 20 but it is not obvious how to factorise x2 (x + 4) and (x + 5).


RULE


The following formula works to factorise all quadratics of the form ax2


The roots are


x bb ac a


= − ± 2


+ bx + c = 0: 2 − 4


Factorising is the reverse of multiplying. Therefore the factors of x2


+ 9 + 20 to get


150


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