Logics, IT studies maths: Is there ANY example in which (x+y)^2 = x^2 + y^2 ?

[tyrael864]

Hello,
I need to do an exercise for my studies and I do not know whether or not my way of thinking is right.
Is there ANY example in which (x+y)^2 = x^2 + y^2. I am aware that this is incorrect in most cases but what if we put zero in place of x, in that case we would get 0 ^2 + y^2 but I do not know if it still counts as true. I appreciate any help. Thank you.

Edit: Both x and y must belong to R

 

[mmm4444bot]

I need to do an exercise for my studies …

Is there ANY example in which (x+y)^2 = x^2 + y^2 [?]

… what if we put zero in place of x, in that case we would get 0^2 + y^2 but I do not know if it still counts as true.

If you're not sure that x = 0 leads to a true statement, make the substitutions for x and simplify.

([0] + y)^2 = [0]^2 + y^2

This simplifies to:

y^2 = y^2

Now what do you think?

There are two other examples.

 

[tkhunny]

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

You are asking if 2xy = 0. There are only very few ways to accomplish that. There are no "examples," only those very few cases where it is so.


You should also note that \(\displaystyle (x+y)^{2}\ge 0\), so, making it zero is the only way. \(\displaystyle x = -y\)

 

[mmm4444bot]

…There are no "examples," only those very few cases …

Not sure what your quotation denotes, but 'cases' is a synonym of 'examples'. :wink:

 

[tkhunny]

Cases are exhaustive. Examples are not.

Anyway, somehow that is stuck in my head.

 

[stapel]

Is there ANY example in which (x+y)^2 = x^2 + y^2?

You've already seen an explanation of how to check the one case that you'd rumbled upon. Now try using what you've learned in algebra (logic, etc) to approach this. Expanding the left-hand side, you have:

. . . . .\(\displaystyle x^2\, +\, 2xy\, +\, y^2\, =\, x^2\, +\, y^2\)

Subtracting from either side, we get:

. . . . .\(\displaystyle 2xy\, =\, 0\)

Under what conditions is this equation true?