Toán Lớp 8: chứng minh đa thức k âm với mọi x thộc Z : 5x^2+y^2-4xy-2y+8x+2013
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Toán Lớp 8: chứng minh đa thức k âm với mọi x thộc Z : 5x^2+y^2-4xy-2y+8x+2013
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5{x^2} + {y^2} – 4xy – 2y + 8x + 2013\\
\Rightarrow (4{x^2} – 4xy + {y^2}) + (4x – 2y) + ({x^2} + 4x + 4) + 2009\\
\Rightarrow {(2x – y)^2} + 2(2x – y) + 1 + {(x + 2)^2} + 2008\\
\Rightarrow {(2x – y + 1)^2} + {(x + 2)^2} + 2008 > 0\forall x
\end{array}\]
5{x^2} + {y^2} – 4xy – 2y + 8x + 2013\\
\Rightarrow (4{x^2} – 4xy + {y^2}) + (4x – 2y) + ({x^2} + 4x + 4) + 2009\\
\Rightarrow {(2x – y)^2} + 2(2x – y) + 1 + {(x + 2)^2} + 2008\\
\Rightarrow {(2x – y + 1)^2} + {(x + 2)^2} + 2008 > 0\forall x
\end{array}\]