Bunuel wrote:
OFFICIAL SOLUTION:
If x and y are integers and \(-x \leq y \leq x\), does \(\sqrt{x^2 - y^2} = x + y\)?
First of all,\(-x \leq y \leq x\) ensures two things:
1.\(x^2-y^2\geq 0\) , so the square root of this number will be defined.
2.\(x+y\geq 0\) , so the square root won't be equal to negative number.
Next,\(-x \leq x\) implies that\(x \geq 0\) .
And finally, before moving to the statements, let's rephrase the question:
Does\(\sqrt{x^2 - y^2} = x + y\) ?
Square both sides: does\(x^2 - y^2 = x^2+2xy + y^2\) ?
Does\(xy+y^2=0\) ?Notice here that we cannot reduce this by y, because we'll loose a possible root: \(y=0\).
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