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a) điều kiện \(x;y\ge0\) ; \(x\ne y\)
b) ta có : \(A=\dfrac{\left(\sqrt{x}-\sqrt{y}\right)^2+4\sqrt{xy}}{\sqrt{x}+\sqrt{y}}-\dfrac{x-y}{\sqrt{x}-\sqrt{y}}\)
\(\Leftrightarrow A=\dfrac{\left(\sqrt{x}+\sqrt{y}\right)^2}{\sqrt{x}+\sqrt{y}}-\dfrac{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)}{\sqrt{x}-\sqrt{y}}\)
\(\Leftrightarrow A=\left(\sqrt{x}+\sqrt{y}\right)-\left(\sqrt{x}+\sqrt{y}\right)=0\)
\(a.ĐKXĐ:\left\{{}\begin{matrix}x\ne y\\x;y>0\end{matrix}\right.\)
\(b.A=\dfrac{\left(\sqrt{x}-\sqrt{y}\right)^2+4\sqrt{xy}}{\sqrt{x}+\sqrt{y}}-\dfrac{x-y}{\sqrt{x}-\sqrt{y}}=\dfrac{x-2\sqrt{xy}+y+4\sqrt{xy}}{\sqrt{x}+\sqrt{y}}-\dfrac{\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)}{\sqrt{x}-\sqrt{y}}=\dfrac{\left(\sqrt{x}+\sqrt{y}\right)^2}{\sqrt{x}+\sqrt{y}}-\sqrt{x}-\sqrt{y}=\sqrt{x}+\sqrt{y}-\sqrt{x}-\sqrt{y}=0\)
\(\dfrac{\left(\sqrt{X}+\sqrt{Y}\right)\left(1+\sqrt{XY}\right)+\left(\sqrt{X}-\sqrt{Y}\right)\left(1-\sqrt{XY}\right)}{1-XY}\cdot\dfrac{1-XY}{1-XY+\sqrt{X}+\sqrt{Y}+2\sqrt{XY}}=\dfrac{\sqrt{X}+X\sqrt{Y}+\sqrt{Y}+Y\sqrt{X}+\sqrt{X}-X\sqrt{Y}-\sqrt{Y}+Y\sqrt{X}}{1-XY}\cdot\dfrac{1-XY}{XY+X+Y+1}=\dfrac{2\sqrt{X}\left(1+Y\right)}{\left(1+Y\right)\left(X+1\right)}=\dfrac{2\sqrt{X}}{X+1}\)
b: Thay \(x=\dfrac{2}{2+\sqrt{3}}=2\left(2-\sqrt{3}\right)=4-2\sqrt{3}\) vào P, ta được:
\(P=\dfrac{2\left(\sqrt{3}-1\right)}{4-2\sqrt{3}+1}=\dfrac{2\sqrt{3}-2}{5-2\sqrt{3}}=\dfrac{6\sqrt{3}+2}{13}\)
2
\(A=\sqrt{1-6x+9x^2}+\sqrt{9x^2-12x+4}\)
A= \(\sqrt{9x^2-6x+1}+\sqrt{9x^2-12x+4}\)
A= \(\sqrt{\left(3x-1\right)^2}+\sqrt{\left(3x-2\right)^2}=\left|3x-1\right|+\left|3x-2\right|\)
ta có |3x-1|+|3x-2|=|3x-1|+|2-3x| ≥ |3x-1+2-3x|=1
=> A ≥ 1
=> Min A =1 khi 1/3 ≤ x ≤ 2/3
a: \(N=\dfrac{x-\sqrt{xy}+y}{\sqrt{x}+\sqrt{y}}:\left(\dfrac{\left(x-y\right)\left(\sqrt{x}+\sqrt{y}\right)-x\sqrt{x}+y\sqrt{y}}{x-y}\right)\)
\(=\dfrac{x-\sqrt{xy}+y}{\sqrt{x}+\sqrt{y}}:\dfrac{x\sqrt{x}+x\sqrt{y}-y\sqrt{x}-y\sqrt{y}-x\sqrt{x}+y\sqrt{y}}{x-y}\)
\(=\dfrac{x-\sqrt{xy}+y}{\sqrt{x}+\sqrt{y}}\cdot\dfrac{x-y}{x\sqrt{y}-y\sqrt{x}}\)
\(=\dfrac{x-\sqrt{xy}+y}{1}\cdot\dfrac{\sqrt{x}-\sqrt{y}}{\sqrt{xy}\left(\sqrt{x}-\sqrt{y}\right)}\)
\(=\dfrac{x-\sqrt{xy}+y}{\sqrt{xy}}\)
b: \(N-1=\dfrac{x-2\sqrt{xy}+y}{\sqrt{xy}}=\dfrac{\left(\sqrt{x}-\sqrt{y}\right)^2}{\sqrt{xy}}>0\)
=>N>1
a: \(=\dfrac{\sqrt{x}+1-\sqrt{x}+1-2\sqrt{x}}{x-1}=\dfrac{-2\left(\sqrt{x}-1\right)}{x-1}=\dfrac{-2}{\sqrt{x}+1}\)
b: \(=\dfrac{\sqrt{x}-x\sqrt{y}-\sqrt{y}+y\sqrt{x}+\sqrt{x}+x\sqrt{y}+\sqrt{y}+y\sqrt{x}}{1-xy}:\left(\dfrac{x+y+2xy+1-xy}{1-xy}\right)\)
\(=\dfrac{2\sqrt{x}+2y\sqrt{x}}{1-xy}\cdot\dfrac{1-xy}{x+y+xy+1}\)
\(=\dfrac{2\sqrt{x}\left(y+1\right)}{\left(y+1\right)\left(x+1\right)}=\dfrac{2\sqrt{x}}{x+1}\)
c: \(=\dfrac{3x+3\sqrt{x}-9+x+2\sqrt{x}-3-x+4}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-1\right)}\)
\(=\dfrac{3x+5\sqrt{x}-8}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-1\right)}=\dfrac{3\sqrt{x}+8}{\sqrt{x}+2}\)
Nếu có thêm điều kiện \(y>1\) thì kết quả là \(\dfrac{1}{x-1}\)
\(A=\left(\dfrac{\sqrt{x}+\sqrt{y}}{\sqrt{x}-\sqrt{y}}-\dfrac{\sqrt{x}-\sqrt{y}}{\sqrt{x}+\sqrt{y}}\right):\dfrac{\sqrt{xy}}{x-y}\)
\(A=\dfrac{\left(\sqrt{x}+\sqrt{y}\right)^2-\left(\sqrt{x}-\sqrt{y}\right)^2}{x-y}:\dfrac{\sqrt{xy}}{x-y}\)
\(A=\dfrac{x+2\sqrt{xy}+y-x+2\sqrt{xy}-y}{\sqrt{xy}}=\dfrac{4\sqrt{xy}}{\sqrt{xy}}=4\)
\(A=\left(\dfrac{\sqrt{x}+\sqrt{y}}{\sqrt{x}-\sqrt{y}}-\dfrac{\sqrt{x}-\sqrt{y}}{\sqrt{x}+\sqrt{y}}\right):\dfrac{\sqrt{xy}}{x-y}\)
\(=\dfrac{\left(\sqrt{x}+\sqrt{y}\right)^2-\left(\sqrt{x}-\sqrt{y}\right)^2}{x-y}\cdot\dfrac{x-y}{\sqrt{xy}}\)
\(=\dfrac{\left(\sqrt{x}+\sqrt{y}+\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}-\sqrt{x}+\sqrt{y}\right)}{x-y}\cdot\dfrac{x-y}{\sqrt{xy}}\)
\(=\dfrac{2\sqrt{x}\cdot2\sqrt{y}}{x-y}\cdot\dfrac{x-y}{\sqrt{xy}}\)
\(=\dfrac{4\sqrt{xy}}{x-y}\cdot\dfrac{x-y}{\sqrt{xy}}\)
\(=4\)
a) tự làm.
b) \(P=\left(\dfrac{x-y}{\sqrt{x}-\sqrt{y}}-\dfrac{x\sqrt{x}-y\sqrt{y}}{x-y}\right)\div\dfrac{\left(\sqrt{x}-\sqrt{y}\right)^2+\sqrt{xy}}{\sqrt{x}+\sqrt{y}}\)
\(=\left(\dfrac{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)}{\sqrt{x}-\sqrt{y}}-\dfrac{\left(\sqrt{x}-\sqrt{y}\right)\left(x+\sqrt{xy}+y\right)}{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)}\right)\cdot\dfrac{\sqrt{x}+\sqrt{y}}{\left(\sqrt{x}-\sqrt{y}\right)^2+\sqrt{xy}}\)
\(=\left(\sqrt{x}+\sqrt{y}-\dfrac{x+\sqrt{xy}+y}{\sqrt{x}+\sqrt{y}}\right)\cdot\dfrac{\sqrt{x}+\sqrt{y}}{\left(\sqrt{x}-\sqrt{y}\right)^2+\sqrt{xy}}\)
\(=\dfrac{\sqrt{x}\left(\sqrt{x}+\sqrt{y}\right)+\sqrt{y}\left(\sqrt{x}+\sqrt{y}\right)-\left(x+\sqrt{xy}+y\right)}{\sqrt{x}+\sqrt{y}}\cdot\dfrac{\sqrt{x}+\sqrt{y}}{\left(\sqrt{x}-\sqrt{y}\right)^2+\sqrt{xy}}\)
\(=\dfrac{x+\sqrt{xy}+\sqrt{xy}+y-x-\sqrt{xy}-y}{\sqrt{x}+\sqrt{y}}\cdot\dfrac{\sqrt{x}+\sqrt{y}}{\left(\sqrt{x}-\sqrt{y}\right)^2+\sqrt{xy}}\)
\(=\dfrac{\sqrt{xy}}{\sqrt{x}+\sqrt{y}}\cdot\dfrac{\sqrt{x}+\sqrt{y}}{\left(\sqrt{x}-\sqrt{y}\right)^2+\sqrt{xy}}\)
\(=\sqrt{xy}\cdot\dfrac{1}{\left(\sqrt{x}-\sqrt{y}\right)^2+\sqrt{xy}}\)
\(=\dfrac{\sqrt{xy}}{x-2\sqrt{xy}+y+\sqrt{xy}}\)
\(=\dfrac{\sqrt{xy}}{x-\sqrt{xy}+y}\)