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Simplify each of the following expressions:
\( (x+y-2 z)^{2}-x^{2}-y^{2}-3 z^{2}+4 x y \)
Given:
\( (x+y-2 z)^{2}-x^{2}-y^{2}-3 z^{2}+4 x y \)
To do:
We have to simplify \( (x+y-2 z)^{2}-x^{2}-y^{2}-3 z^{2}+4 x y \).
Solution:
We know that,
$(a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca$
Therefore,
$(x+y-2 z)^{2}-x^{2}-y^{2}-3 z^{2}+4 x y=x^{2}+y^{2}+4 z^{2}+2 x y-4 y z-4 z x-x^{2}-y^{2}-3 z^{2}+4 x y$
$=z^{2}+6 x y-4 y z-4 z x$
Hence, $(x+y-2 z)^{2}-x^{2}-y^{2}-3 z^{2}+4 x y=z^{2}+6 x y-4 y z-4 z x$.
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