1.
\(Q=\dfrac{1}{x^2+y^2}+\dfrac{1}{2xy}+\left(4xy+\dfrac{1}{4xy}\right)+\dfrac{5}{4xy}+2016\)
\(Q\ge\dfrac{4}{x^2+y^2+2xy}+2\sqrt{\dfrac{4xy}{4xy}}+\dfrac{5}{\left(x+y\right)^2}+2016=2027\)
\(Q_{min}=2027\) khi \(x=y=\dfrac{1}{2}\)
2a.
ĐKXĐ: \(x\ge1\)
Đặt \(\left\{{}\begin{matrix}\sqrt[3]{2-x}=a\\\sqrt{x-1}=b\ge0\end{matrix}\right.\) \(\Rightarrow a^3+b^2=1\)
Ta được hệ:
\(\left\{{}\begin{matrix}a+b=1\\a^3+b^2=1\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}a=1-b\\a^3+b^2=1\end{matrix}\right.\)
\(\Leftrightarrow\left(1-b\right)^3+b^2=1\)
\(\Leftrightarrow b^3-4b^2+3b=0\Rightarrow\left[{}\begin{matrix}b=0\\b=1\\b=3\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}x=1\\x=2\\x=10\end{matrix}\right.\)
2b.
\(xy+x+y=x^2-2y^2\)
\(\Leftrightarrow x^2-xy-y^2-\left(x+y\right)=0\)
\(\Leftrightarrow\left(x+y\right)\left(x-2y\right)-\left(x+y\right)=0\)
\(\Leftrightarrow\left(x+y\right)\left(x-2y-1\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x=-y\\x=2y+1\end{matrix}\right.\)
Thay xuống pt dưới:
\(\left[{}\begin{matrix}y^2-y^2-3\left(-y\right)+2y-10=0\\\left(2y+1\right)^2-y^2-3\left(2y+1\right)+2y-10=0\end{matrix}\right.\)
\(\Leftrightarrow...\)
