SOLUTION: the product of two consecutive even numbers is 2208.what is the greatest number? (note:i want short cut method)

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Question 695852: the product of two consecutive even numbers is 2208.what is the greatest number?
(note:i want short cut method)
Found 2 solutions by mouk, mananth:
Answer by mouk(232) About Me  (Show Source):
You can put this solution on YOUR website!
Suppose the two consecutive even numbers are 2n and 2n+2
+%282n%29%282n%2B2%29+=+2208+
+%282%29%28n%29%282%29%28n%2B1%29+=+2208+
+%284%29%28n%29%28n%2B1%29+=+2208+
+%28n%29%28n%2B1%29+=+552+
+n%5E2%2Bn+=+552+
+n%5E2%2Bn+-+552+=+0+
+%28n-23%29%28n%2B24%29+=+0+
So +n=23+ or +n=-24+


When n=23:
The two consecutive even numbers would be +2n+ and +2n%2B2+
ie 46 and 48 (whose product is 2208)


When n=-24:
The two consecutive even numbers would be +2n+ and +2n%2B2+
ie -48 and -46 (whose product is 2208)


So to answer the question, greatest number is 48

Answer by mananth(16946) About Me  (Show Source):
You can put this solution on YOUR website!
x(x+ 2)=2208
x^+2 x +1=2208+1
(x+1)^2 =2209
take the square root
x+1=+/- 47
x= 46
46,48
or
-48,-46