What Is The Formula For (X Y Z) ^2? Maths Q&A

     
Your logic is correct for positive solutions. Now you need the sign pattern khổng lồ be +++,+--,-+-,--+, four possibilities, so multiply by four.

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Applying the AM-GM is the right strategy, but you need to bởi it a bit differently. X^2+4xy+4y^2+4z^2 = x^2+2xy+2xy+4y^2+2z^2+2z^2 ge 6sqrt<6>x^2 cdot 2xy cdot 2xy cdot 4y^2 cdot 2z^2 cdot 2z^2 ...
First solve q(x,y,z) = 2 gian lận 8, then use Hensel's Lemma khổng lồ lift lớn a solution in mathbbZ_2. When working with quadratic forms in characteristic 2, usually it helps lớn work mod 8. A good reason ...

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You can ask WolframAlpha , if you want, but in this case, you can vày it also by hand: Multiply the first three equations by x, y resp. Z, get < 27m = 2xy + 2xz = 2xy+yz = 2xz + yz > So 2xy = 2xz ...
Use Newton-Girard relations between elementary symmetric finctions and sums of powers: Let p_k=x^k+y^k+z^k, ,s_1=x+y+z, ,s_2=xy+yz+zx, ,s_3=xyz. We have: eginalign* p_1&=s_1& ...

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This is intended as a complement to Alex Bartel's answer. (1) Consider the 2d case given by x = x(t), y = y(t). Here is a graph of a parametrized function with (x(t_0),y(t_0)) (in red) và (x(t_0 + delta), y(t_0 + delta)) ...
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left< eginarray l l 2 & 3 \ 5 & 4 endarray ight> left< eginarray l l l 2 và 0 & 3 \ -1 & 1 & 5 endarray ight>
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