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Simpler way to find squares by hand?

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You could take any number and square it on a calculator and find its value. If you do not have means to find the square with technology it is still generally simple to manually compute such a square. However, there are some instances when such a square may be challenging. When this situation arises, there is a simple and straightforward approach:

let x be the number you wish to square then

x - 25 = the hundreds value

(50 - x)^2 = the ones value

add them together and you have your square. It is that simple.

Examples:

46^2 =

46-25 = 21 [hundreds] AND (50-46)^2= 4^2=16, so 46^2 = 2116

43^2=

43-25 = 18 AND (50-43)^2 = 7^2=49, so 43^2 = 1849

and just to show you a silly example as further evidence that it works...

4^2 =

4-25= -21 AND (50-4)^2= 46^2 = 2116 and -2100 + 2116 = 16 !! so 4^2=16

Now your task is to either prove this true for all real numbers or find 1 counter example. If a counter example is found, then for what numbers is this true?

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(x-25)*100 + (50-x)*(50-x) = 100*x - 2500 + 2500 - 50*x - 50*x + x^2 = x^2

This means that your method works perfectly for every real number x.

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