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\(\frac{x+2}{x\sqrt{x}-1}+\frac{\sqrt{x}+1}{x\sqrt{x}-1}-\frac{1}{\sqrt{x-1}}\)
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H9
HT.Phong (9A5)
CTVHS
5 tháng 8 2023
\(A=\left(\dfrac{1}{\sqrt{x-1}}+\dfrac{1}{\sqrt{x-1}}\right)^2\cdot\dfrac{x^2-1}{2}-\sqrt{x^2-1}\) (ĐK: \(x>1\))
\(A=\left(\dfrac{2}{\sqrt{x-1}}\right)^2\cdot\dfrac{x^2-1}{2}-\sqrt{x^2-1}\)
\(A=\dfrac{4}{x-1}\cdot\dfrac{\left(x+1\right)\left(x-1\right)}{2}-\sqrt{x^2-1}\)
\(A=2\left(x+1\right)-\sqrt{\left(x+1\right)\left(x-1\right)}\)
\(A=\sqrt{x+1}\left(2\sqrt{x+1}-\sqrt{x-1}\right)\)
HQ
Hà Quang Minh
Giáo viên
5 tháng 8 2023
\(A=\left(\dfrac{1}{\sqrt{x-1}}+\dfrac{1}{\sqrt{x+1}}\right)^2\cdot\dfrac{x^2-1}{2}-\sqrt{x^2-1}\\ \Rightarrow A=\left(\dfrac{\sqrt{x+1}+\sqrt{x-1}}{\sqrt{x^2-1}}\right)^2\cdot\dfrac{x^2-1}{2}-\sqrt{x^2-1}\\ \Rightarrow A=\dfrac{\left(\sqrt{x+1}+\sqrt{x-1}\right)^2}{2}-\sqrt{x^2-1}\\ \Rightarrow A=\dfrac{2x+2\sqrt{x^2-1}-2\sqrt{x^2-1}}{2}\\ \Rightarrow A=x\)
29 tháng 3 2020
Ta có : A = \(\left(\frac{x+2}{x.\sqrt{x}-1}+\frac{\sqrt{x}+2}{x+\sqrt{x}+1}+\frac{1}{1-\sqrt{x}}\right):\frac{\sqrt{x}+1}{x+\sqrt{x}+1}\)
= \(\frac{x+2+x+\sqrt{x}-2-x-\sqrt{x}-1}{\left(\sqrt{x}-1\right).\left(x+\sqrt{x}+1\right)}.\frac{x+\sqrt{x}+1}{\sqrt{x}+1}\)
= \(\frac{x-1}{\left(\sqrt{x}-1\right).\left(x+\sqrt{x}+1\right)}.\frac{x+\sqrt{x}+1}{\sqrt{x}+1}=1\)
Vậy A = 1
S
12 tháng 6 2019
\(=\left(\frac{x}{2\sqrt{x}}-\frac{1}{2\sqrt{x}}\right)\left(\frac{x\sqrt{x}}{\sqrt{x}+1}-\frac{x+\sqrt{x}}{\sqrt{x}-1}\right)\)
\(=\left(\frac{x-1}{2\sqrt{x}}\right)\left(\frac{x\sqrt{x}\left(\sqrt{x}-1\right)}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}-\frac{\left(x+\sqrt{x}\right)\left(\sqrt{x}+1\right)}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\right)\)
\(=\left(\frac{x-1}{2\sqrt{x}}\right)\left(\frac{x^2-x\sqrt{x}}{x-1}-\frac{x\sqrt{x}+2x+\sqrt{x}}{x-1}\right)\)
\(=\left(\frac{x-1}{2\sqrt{x}}\right)\left(\frac{x^2-2x\sqrt{x}-2x-\sqrt{x}}{x-1}\right)=\frac{x^2-\sqrt{x}-2x\sqrt{x}-2x}{2\sqrt{x}}=\frac{x\sqrt{x}-1-2x-2\sqrt{x}}{2}\)
12 tháng 6 2019
\(\left(\frac{\sqrt{x}}{2}-\frac{1}{2\sqrt{x}}\right)\left(\frac{x\sqrt{x}}{\sqrt{x}+1}-\frac{x+\sqrt{x}}{\sqrt{x}-1}\right)\)
\(=\frac{x-1}{2\sqrt{x}}.\frac{x\sqrt{x}\left(\sqrt{x}-1\right)-\left(x+\sqrt{x}\right)\left(\sqrt{x}+1\right)}{x-1}\)
\(=\frac{x^2-x\sqrt{x}-\left(x\sqrt{x}+x+x+\sqrt{x}\right)}{2\sqrt{x}}\)
\(=\frac{x^2-x\sqrt{x}-x\sqrt{x}-2x-\sqrt{x}}{2\sqrt{x}}\)
\(=\frac{x^2-2x\sqrt{x}-2x-\sqrt{x}}{2\sqrt{x}}\)
29 tháng 6 2019
ĐK : x>0, x khác 1
\(A=\left(\frac{1}{\sqrt{x}+1}+\frac{2\left(1-\sqrt{x}\right)}{x\left(\sqrt{x}+1\right)-\left(\sqrt{x}+1\right)}\right):\left(\frac{\sqrt{x}+2}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-1\right)}-\frac{2}{x-1}\right)\)
\(=\left(\frac{1}{\sqrt{x}+1}-\frac{2\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)^2}\right):\left(\frac{1}{\sqrt{x}-1}-\frac{2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\right)\)
\(=\frac{\sqrt{x}+1-2}{\left(\sqrt{x}+1\right)^2}:\frac{\sqrt{x}+1-2}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}=\frac{\sqrt{x}-1}{\left(\sqrt{x}+1\right)^2}.\frac{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}{\sqrt{x}-1}\)
\(=\frac{\sqrt{x}-1}{\sqrt{x}+1}\)
10 tháng 7 2019
\(A=\frac{\sqrt{x}+1}{x-1}-\frac{x+2}{x\sqrt{x}-1}-\frac{\sqrt{x}+1}{x+\sqrt{x}+1}\)
\(=\frac{\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)\(-\frac{x-2}{\sqrt{x}^3-1}\)\(-\frac{\sqrt{x}+1}{x+\sqrt{x}+1}\)
\(=\frac{1}{\sqrt{x}-1}\)\(-\frac{x-2}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\)\(-\frac{\sqrt{x}+1}{x+\sqrt{x}+1}\)
\(=\frac{x+\sqrt{x}+1-x+2-\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\)
\(=\frac{2}{\sqrt{x}^3-1}\)