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1/
\(x^2-xy-2y^2=0\Leftrightarrow x^2+xy-2xy-2y^2=0\)
\(\Leftrightarrow x\left(x+y\right)-2y\left(x+y\right)=0\)
\(\Leftrightarrow\left(x+y\right)\left(x-2y\right)=0\Rightarrow x=2y\) (do \(x+y\ne0\))
\(\Rightarrow P=\frac{2y-y}{2y+y}=\frac{y}{3y}=\frac{1}{3}\)
2/
\(x^4-30x^2+31x-30=0\)
\(\Leftrightarrow x^4+x-30x^2+30x-30=0\)
\(\Leftrightarrow x\left(x^3+1\right)-30\left(x^2-x+1\right)=0\)
\(\Leftrightarrow x\left(x+1\right)\left(x^2-x+1\right)-30\left(x^2-x+1\right)=0\)
\(\Leftrightarrow\left(x^2+x-30\right)\left(x^2-x+1\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x^2+x-30=0\\x^2-x+1=0\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}\left(x-5\right)\left(x+6\right)=0\\\left(x-\frac{1}{2}\right)^2+\frac{3}{4}=0\left(vn\right)\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}x=5\\x=-6\end{matrix}\right.\)
\(x+y=1\Rightarrow\left\{{}\begin{matrix}y-1=-x\\x-1=-y\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}\left(y-1\right)^2=x^2\\\left(x-1\right)^2=y^2\end{matrix}\right.\)
\(\frac{x}{y^3-1}-\frac{y}{x^3-1}+\frac{2\left(x-y\right)}{\left(xy\right)^2+3}=\frac{x}{\left(y-1\right)\left(y^2+y+1\right)}-\frac{y}{\left(x-1\right)\left(x^2+x+1\right)}+\frac{2\left(x-y\right)}{\left(xy\right)^2+3}\)
\(=\frac{-1}{y^2+y+1}+\frac{1}{x^2+x+1}+\frac{2\left(x-y\right)}{\left(xy\right)^2+3}=\frac{-1}{x^2+3y}+\frac{1}{y^2+3x}+\frac{2\left(x-y\right)}{\left(xy\right)^2+3}\)
\(=\frac{-y^2-3x+x^2+3y}{\left(xy\right)^2+3x^3+3y^3+9xy}+\frac{2\left(x-y\right)}{\left(xy\right)^2+3}=\frac{\left(x-y\right)\left(x+y\right)-3x+3y}{\left(xy\right)^2+3\left(x+y\right)\left(\left(x+y\right)^2-3xy\right)+9xy}+\frac{2\left(x-y\right)}{\left(xy\right)^2+3}\)
\(=\frac{-2\left(x-y\right)}{\left(xy\right)^2+3}+\frac{2\left(x-y\right)}{\left(xy\right)^2+3}=0\)
6) Ta có
\(A=\frac{x^3}{y+2z}+\frac{y^3}{z+2x}+\frac{z^3}{x+2y}\)
\(=\frac{x^4}{xy+2xz}+\frac{y^4}{yz+2xy}+\frac{z^4}{zx+2yz}\)
\(\ge\frac{\left(x^2+y^2+z^2\right)^2}{xy+2xz+yz+2xy+zx+2yz}\)
\(\Leftrightarrow A\ge\frac{1}{3\left(xy+yz+zx\right)}\ge\frac{1}{3\left(x^2+y^2+z^2\right)}=\frac{1}{3}\)