\(x^2-2x+3=t\left(t\ge0\right)\)

\(pt\Leftrightarrow\frac{1}{t-1}+\frac{1}{t}=\frac{9}{2\left(t+1\right)}\)

\(\Leftrightarrow\frac{2t\left(t+1\right)}{2t\left(t^2-1\right)}+\frac{2\left(t^2-1\right)}{2t\left(t^2-1\right)}-\frac{9t\left(t-1\right)}{2t\left(t^2-1\right)}=0\)

\(\Leftrightarrow-5t^2+11t-2=0\)

\(\Leftrightarrow\left(5x-1\right)\left(x-2\right)=0\)

\(\Leftrightarrow\orbr{\begin{cases}x=\frac{1}{5}\\x=2\end{cases}}\)

\(=\left(\frac{\sqrt{y}}{\sqrt{x}\left(\sqrt{x}+\sqrt{y}\right)}-\frac{\sqrt{x}}{\sqrt{y}\left(\sqrt{x}+\sqrt{y}\right)}\right)\cdot\frac{\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)}{2\sqrt{xy}}\)

\(=\left(\frac{y-x}{\sqrt{xy}\left(\sqrt{x}+\sqrt{y}\right)}\right)\cdot\frac{\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)}{2\sqrt{xy}}\)

\(=\frac{\left(y-x\right)\left(\sqrt{x}-\sqrt{y}\right)}{2xy}\)

\(\left(\frac{\sqrt{y}}{x+\sqrt{xy}}-\frac{\sqrt{y}}{x-\sqrt{xy}}\right).\frac{x-y}{2\sqrt{xy}}\)

\(=\frac{x-y}{2\sqrt{xy}}.\left[-\frac{2y\sqrt{x}}{\left(x+\sqrt{yx}\right)\left(x-\sqrt{yx}\right)}\right]\)

\(=-\frac{2y\sqrt{x}}{\left(x+\sqrt{xy}\right)\left(x-\sqrt{xy}\right)}.\frac{x-y}{2\sqrt{xy}}\)

\(=-\frac{2y\sqrt{x}.\left(x-y\right)}{\left(x+\sqrt{xy}\right)\left(x-\sqrt{xy}\right).2\sqrt{xy}}\)

\(=-\frac{y\sqrt{x}\left(x-y\right)}{\left(x+\sqrt{xy}\right)\left(x-\sqrt{xy}\right).\sqrt{xy}}\)

\(=-\frac{\sqrt{y}\left(x-y\right)}{\left(x+\sqrt{yx}\right)\left(x-\sqrt{yx}\right)}\)

\(=-\frac{\sqrt{y}}{x}\)

Đặt \(\left(\sqrt{a};\sqrt{b};\sqrt{c}\right)\rightarrow\left(x;y;z\right)\)\(\Rightarrow\)\(x^2+y^2+z^2=4\)

\(P=\frac{x^3}{x+3y}+\frac{y^3}{y+3z}+\frac{z^3}{z+3x}=\frac{x^4}{x^2+3xy}+\frac{y^4}{y^2+3yz}+\frac{z^4}{z^2+3zx}\)

\(\ge\frac{\left(x^2+y^2+z^2\right)^2}{x^2+y^2+z^2+3\left(xy+yz+zx\right)}\ge\frac{\left(x^2+y^2+z^2\right)^2}{x^2+y^2+z^2+3\left(x^2+y^2+z^2\right)}=\frac{4^2}{4+3.4}=1\)

Dấu "=" xảy ra \(\Leftrightarrow\)\(a=b=c=\frac{2}{\sqrt{3}}\)

à nhầm, \(a=b=c=\frac{4}{3}\) nhé 

\(2x^3-3x^2+2x-3\)

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

\(=\left(x^2+1\right)\left(2x-3\right)\)

\(x^2-2xy+y^2-16\)

\(=\left(x-y\right)^2-4^2\)

\(=\left(x-y-4\right)\left(x-y+4\right)\)

\(2x^3-3x^2+2x-3\)

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

\(=2x\left(x^2+1\right)-3\left(x^2+1\right)\)

\(=\left(x^2+1\right)\left(2x-3\right)\)

\(3\times y^2-12\times y+12\times\)

\(=3\times\cdot\left(y^2-4y+4\right)\)

\(=3\times\cdot\left(y^2-2\cdot2y+2^2\right)\)

\(=3\times\cdot\left(y-2\right)^2\)

\(3xy^2-12xy+12x\)

\(=3x\left(y^2-4y+4\right)\)

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