Question

Cho P = xy(x - 2)(y + 6) + 12x² - 24x + 3y² + 18y + 36. Chứng tỏ P > 0 với mọi x, y thuộc R.

Answer 1

\(P=\left(x^2-2x\right)\left(y^2+6y\right)+12\left(x^2-2x\right)+3\left(y^2+6y\right)+36\)

\(=\left(x^2-2x\right)\left(y^2+6y+12\right)+3\left(y^2+6y+12\right)\)

\(=\left(x^2-2x+3\right)\left(y^2+6y+12\right)\)

\(=\left[\left(x-1\right)^2+2\right]\left[\left(y+3\right)^2+3\right]>0\)

Answer 2

\(P=xy\left(x-2\right)\left(x+6\right)+12x^2-24x+3y^2+18y+36\)

\(=xy\left(x-2\right)\left(x+6\right)+12x\left(x-2\right)+3y\left(y+6\right)+36\)

Đặt \(\left\{{}\begin{matrix}x-2=a\\x+6=b\end{matrix}\right.\) . Khi đó

\(P=xy.a.b+12x.a+3y.b+36\)

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