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25 tháng 2 2022
\(A=\left(\dfrac{\sqrt{x}}{\sqrt{x}-1}+\dfrac{1}{x-\sqrt{x}}\right):\left(\dfrac{1}{\sqrt{x}+1}+\dfrac{2}{x-1}\right)\)
\(\Rightarrow A=\left(\dfrac{x}{\sqrt{x}\left(\sqrt{x}-1\right)}+\dfrac{1}{\sqrt{x}\left(\sqrt{x}-1\right)}\right):\left(\dfrac{\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}+\dfrac{2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\right)\)
\(\Rightarrow A=\dfrac{x+1}{\sqrt{x}\left(\sqrt{x}-1\right)}:\dfrac{\sqrt{x}-1+2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)
\(\Rightarrow A=\dfrac{x+1}{\sqrt{x}\left(\sqrt{x}-1\right)}:\dfrac{\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)
\(\Rightarrow A=\dfrac{x+1}{\sqrt{x}\left(\sqrt{x}-1\right)}.\dfrac{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{\sqrt{x}+1}\)
\(\Rightarrow A=\dfrac{x+1}{\sqrt{x}}\)
CN
1
NH
0
NV
Nguyễn Việt Lâm
Giáo viên
21 tháng 1 2024
\(Q=\left(\dfrac{1}{2\sqrt{x}+1}+\dfrac{1}{2\sqrt{x}-1}\right):\dfrac{1}{1-4x}\)
\(=\left(\dfrac{2\sqrt{x}-1}{\left(2\sqrt{x}+1\right)\left(2\sqrt{x}-1\right)}+\dfrac{2\sqrt{x}+1}{\left(2\sqrt{x}+1\right)\left(2\sqrt{x}-1\right)}\right).\left(1-4x\right)\)
\(=\left(\dfrac{2\sqrt{x}-1+2\sqrt{x}+1}{4x-1}\right)\left(1-4x\right)\)
\(=\dfrac{-4\sqrt{x}.\left(4x-1\right)}{4x-1}=-4\sqrt{x}\)
T
21 tháng 1 2024
\(Q=\left(\dfrac{1}{2\sqrt{x}+1}+\dfrac{1}{2\sqrt{x}-1}\right):\dfrac{1}{1-4x}\left(dkxd:x\ge0;x\ne\dfrac{1}{4}\right)\)
\(=\left[\dfrac{2\sqrt{x}-1}{\left(2\sqrt{x}-1\right)\left(2\sqrt{x}+1\right)}+\dfrac{2\sqrt{x}+1}{\left(2\sqrt{x}-1\right)\left(2\sqrt{x}+1\right)}\right]\cdot\left(1-4x\right)\)
\(=\dfrac{2\sqrt{x}-1+2\sqrt{x}+1}{4x-1}\cdot\left[-\left(4x-1\right)\right]\)
\(=4\sqrt{x}\cdot\left(-1\right)\)
\(=-4\sqrt{x}\)
HP
12 tháng 9 2021
ĐK: \(x\ge0;x\ne4;x\ne9\)
\(Q=\left(\dfrac{x-3\sqrt{x}}{9-x}+1\right):\left(\dfrac{9-x}{x+\sqrt{x}-6}+\dfrac{\sqrt{x}-3}{\sqrt{x}-2}+\dfrac{2-\sqrt{x}}{\sqrt{x}+3}\right)\)
\(=\left[-\dfrac{\sqrt{x}\left(\sqrt{x}-3\right)}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}+1\right]:\left[\dfrac{9-x}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-2\right)}+\dfrac{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-2\right)}+\dfrac{\left(2-\sqrt{x}\right)\left(\sqrt{x}-2\right)}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-2\right)}\right]\)
\(=\dfrac{3}{\sqrt{x}+3}:\dfrac{-\left(\sqrt{x}-2\right)^2}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-2\right)}\)
\(=\dfrac{3}{\sqrt{x}+3}.\dfrac{-\left(\sqrt{x}+3\right)}{\sqrt{x}-2}\)
\(=\dfrac{3}{2-\sqrt{x}}\)
NV
3
HH
13 tháng 12 2020
a,b) Đk để biểu thức A xác định là x > 4
\(A=\frac{x\left(\sqrt{\left(\sqrt{x-4}+2\right)^2}+\sqrt{\left(\sqrt{x-4}-2\right)^2}\right)}{\sqrt{\left(x-4\right)^2}}\)
\(A=\frac{x\left(|\sqrt{x-4}+2|+|\sqrt{x-4}-2|\right)}{|x-4|}\)
\(A=\frac{x\left(\sqrt{x-4}+2+|\sqrt{x-4}-2|\right)}{x-4}\)
+) Nếu 4 < x < 8 thì \(\sqrt{x-4}-2< 0\)nên \(A=\frac{x\left(\sqrt{x-4}+2+2-\sqrt{x-4}\right)}{x-4}=\frac{4x}{x-4}=4+\frac{16}{x-4}\)
Do 4 < x < 8 nên 0 < x - 4 < 4 => A > 88
+) Nếu \(x\ge8\)thì \(\sqrt{x-4}-2\ge0\)nên :
\(A=\frac{x\left(\sqrt{x-4}+2+\sqrt{x-4}-2\right)}{x-4}=\frac{2x\sqrt{x-4}}{x-4}=\frac{2x}{\sqrt{x-4}}=2\sqrt{x-4}+\frac{8}{\sqrt{x-4}}\ge2\sqrt{16}=8\)
( Theo bđt Cô si )
- Dấu " = " xảy ra khi và chỉ khi \(2\sqrt{x-4}=\frac{8}{\sqrt{x-4}}\Leftrightarrow x-4=4\Leftrightarrow x=8\)
Vậy Min của A = 8 khi x = 8
c) Xét 4 < x < 8 thì \(A=4+\frac{16}{x-4}\), ta thấy \(A\in Z\)khi và chỉ khi \(\frac{16}{x-4}\in Z\Leftrightarrow x-4\)là ước nguyên dương của 16
- Hay \(x-4\in\left\{1;2;4;16\right\}\Leftrightarrow x=\left\{5;6;8;12;20\right\}\)đối chiếu điều kiện => x = 5 hoặc x = 6
+) Xét \(x\ge8\)ta có : \(A=\frac{2x}{\sqrt{x-4}}\)
Đặt \(\sqrt{x-4}=m\Rightarrow\hept{\begin{cases}x=m^2+4\\m\ge2\end{cases}}\)khi đó ta có : \(A=\frac{2\left(m^2+4\right)}{m}=2m+\frac{8}{m}\)
\(\Rightarrow m\in\left\{2;4;8\right\}\Leftrightarrow x\in\left\{8;20;68\right\}\)
Vậy để A nhận giá trị nguyên thì \(x\in\left\{5;6;8;20;68\right\}\)
Với `x >= 0,x ne 1` có:
Bth`=[2x+4+(\sqrt{x}+2)(\sqrt{x}-1)-2(x+\sqrt{x}+1)]/[(\sqrt{x}-1)(x+\sqrt{x}+1)]`
`=[2x+4+x-\sqrt{x}+2\sqrt{x}-2-2x-2\sqrt{x}-2]/[(\sqrt{x}-1)(x+\sqrt{x}+1)]`
`=[x-\sqrt{x}]/[(\sqrt{x}-1)(x+\sqrt{x}+1)]`
`=[\sqrt{x}(\sqrt{x}-1)]/[(\sqrt{x}-1)(x+\sqrt{x}+1)]`
`=\sqrt{x}/[x+\sqrt{x}+1]`
= \(\dfrac{2x+4}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\) + \(\dfrac{\left(\sqrt{x}+2\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\) - \(\dfrac{2x+2\sqrt{x}+2}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\)
= \(\dfrac{2x+4+x-\sqrt{x}+2\sqrt{x}-2-2x-2\sqrt{x}-2}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\)
= \(\dfrac{x-\sqrt{x}}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\)
= \(\dfrac{\sqrt{x}}{x+\sqrt{x}+1}\)