Determine whether the property is true for all integers, true for no integers, true for only some integers.

?(x+y)?^2 = x^2 + y^2

?(x+y)?^2 = x^2 + y^2

?(x+y)?^2 = x^2 + y^2

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Tarzana, CA

Daniel,

I'm not sure what the question marks represent in your problem statement ?(x+y)?^2 = x^2 + y^2 , so I will ignore them in my solution. If this is incorrect, please restate the problem for me. Thanks.

So the relation would be (x + y)^{2} = x^{2} + y^{2} if the question marks are removed. If we are talking about integers: .. -2, -1, 0, 1, 2, ... then

by cross-multiplying we get x^{2} + 2xy + y^{2} = x^{2} + y^{2}, which would mean 2xy = 0, so x = 0 or y = 0. So the statement is true if either x or y is equal to zero, and otherwise it's false.

But let's explore this a bit further.

Suppose you were working on a number system that was like the integers but not exactly - say on a 6-hour clock. So that, for example, 2*3 = 6
*which is equal to 0 on a 6-hour clock*. So on this 6-hour clock, if x = 2 and y = 3, then xy = 0 and we would have (x + y)^{2} = x^{2} + y^{2 }where neither x nor y was zero. So it really depends which number system you are working on whether an equality like this holds.

Woodland Hills, CA

This problem can be approached geometrically:

a^2 + b^2 = c^2

this relationship is Pythagoras theorem of a right triangle with legs sizes a,b and hypotenuse size c

Taking square of both sides.

√( a^2 + b^2) = c

now, if we let √a^2 + b^2 equals a+ b , we are violating the Pythagoras law, implying that

length of the hypotenuse equals sum of the 2 length, which is not true.

Summation of squares being equal square of some quantity does not imply that summation of

length is equal

√ ( 4^2 + 3^2 ) = √( 16 +9 ) = √25 = 5

√4^2 + √3^2 = 4 + 3 = 7

√( a^2 + b^2) ≠ a +b, then

a^2 + b^2 ≠ ( a + b) ^2

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