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a: \(A=\dfrac{\sqrt{x}-1+\sqrt{x}}{\sqrt{x}\left(1-\sqrt{x}\right)}:\left(\dfrac{-\left(2x+\sqrt{x}-1\right)\left(x-\sqrt{x}+1\right)+\left(2x\sqrt{x}+x-\sqrt{x}\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)\left(x-\sqrt{x}+1\right)}\right)\)
\(=\dfrac{2\sqrt{x}-1}{\sqrt{x}\left(1-\sqrt{x}\right)}:\dfrac{-2x^2+x\sqrt{x}-2\sqrt{x}+1+2x^2-x\sqrt{x}-2x+\sqrt{x}}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)\left(x-\sqrt{x}+1\right)}\)
\(=\dfrac{-\left(2\sqrt{x}-1\right)}{\sqrt{x}\left(\sqrt{x}-1\right)}\cdot\dfrac{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)\left(x-\sqrt{x}+1\right)}{-2x-\sqrt{x}+1}\)
\(=\dfrac{-\left(2\sqrt{x}-1\right)\left(\sqrt{x}+1\right)\left(x-\sqrt{x}+1\right)}{-\sqrt{x}\left(2x+\sqrt{x}-1\right)}\)
\(=\dfrac{\left(2\sqrt{x}-1\right)\left(\sqrt{x}+1\right)\left(x-\sqrt{x}+1\right)}{\sqrt{x}\left(\sqrt{x}+1\right)\left(2\sqrt{x}-1\right)}\)
\(=\dfrac{x-\sqrt{x}+1}{\sqrt{x}}\)
b: Thay \(x=17-12\sqrt{2}=\left(3-2\sqrt{2}\right)^2\) vào A, ta được:
\(A=\dfrac{17-12\sqrt{2}-\sqrt{2}+1+1}{3-2\sqrt{2}}=\dfrac{19-13\sqrt{2}}{3-2\sqrt{2}}=5-\sqrt{2}\)
ĐKXĐ : \(\hept{\begin{cases}x\ge0\\x\ne1\\x\ne9\end{cases}}\)
a) \(=\left(\frac{x+2\sqrt{x}-7}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}+\frac{1-\sqrt{x}}{\sqrt{x}-3}\right)\div\left(\frac{\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+3\right)}-\frac{\sqrt{x}+3}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+3\right)}\right)\)
\(=\left(\frac{x+2\sqrt{x}-7}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}+\frac{\left(1-\sqrt{x}\right)\left(\sqrt{x}+3\right)}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\right)\div\left(\frac{\sqrt{x}-1-\sqrt{x}-3}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+3\right)}\right)\)
\(=\left(\frac{x+2\sqrt{x}-7}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}+\frac{3-2\sqrt{x}-x}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\right)\div\left(\frac{-4}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\right)\)
\(=\left(\frac{x+2\sqrt{x}-7+3-2\sqrt{x}-x}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\right)\div\left(\frac{-4}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+3\right)}\right)\)
\(=\frac{-4}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\times\frac{\left(\sqrt{x}-1\right)\left(\sqrt{x}+3\right)}{-4}\)
\(=\frac{\sqrt{x}-1}{\sqrt{x}-3}\)
b) Để \(P\left(\sqrt{x}-3\right)=\left|x-3\right|\)
=> \(\frac{\sqrt{x}-1}{\sqrt{x}-3}\cdot\left(\sqrt{x}-3\right)=\left|x-3\right|\)(\(\hept{\begin{cases}x\ge0\\x\ne1\\x\ne9\end{cases}}\))
=> \(\sqrt{x}-1=\left|x-3\right|\)
=> \(\orbr{\begin{cases}\sqrt{x}-1=x-3\left(x\ge3\right)\\\sqrt{x}-1=3-x\left(1\le x< 3\right)\end{cases}}\)
=> \(\orbr{\begin{cases}x=4\\x=\frac{9-\sqrt{17}}{2}\end{cases}}\)
c) Em chịu T.T
1. \(VT=\sqrt{7+4\sqrt{3}}-\sqrt{7-4\sqrt{3}}\)
\(=\sqrt{2^2+2.2.\sqrt{3}+\left(\sqrt{3}\right)^2}-\sqrt{2^2-2.2.\sqrt{3}+\left(\sqrt{3}\right)^2}\)
\(=\sqrt{\left(2+\sqrt{3}\right)^2}-\sqrt{\left(2-\sqrt{3}\right)^2}\)
\(=2+\sqrt{3}-2+\sqrt{3}=VP\)
Bài 1.
Ta có : \(\sqrt{7+4\sqrt{3}}-\sqrt{7-4\sqrt{3}}\)
\(=\sqrt{3+4\sqrt{3}+4}-\sqrt{3-4\sqrt{3}+4}\)
\(=\sqrt{\left(\sqrt{3}+2\right)^2}-\sqrt{\left(\sqrt{3}-2\right)^2}\)
\(=\left|\sqrt{3}+2\right|-\left|\sqrt{3}-2\right|\)
\(=\sqrt{3}+2-\left(2-\sqrt{3}\right)\)
\(=\sqrt{3}+2-2+\sqrt{3}=2\sqrt{3}\left(đpcm\right)\)
B3: \(\sqrt{x^4-4x^3+2x^2+4x+1}=3x-1\)
\(pt\Leftrightarrow x^4-4x^3+2x^2+4x+1=\left(3x-1\right)^2\)
\(\Leftrightarrow x^4-4x^3+2x^2+4x+1=9x^2-6x+1\)
\(\Leftrightarrow x^4-4x^3-7x^2+10x=0\)
\(\Leftrightarrow x\left(x^3-4x^2-7x+10\right)=0\)
\(\Leftrightarrow x\left(x-1\right)\left(x-5\right)\left(x+2\right)=0\)
\(\Rightarrow\orbr{\begin{cases}x=1\\x=5\end{cases}}\) (thỏa mãn (mấy cái kia loại hết))