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27 tháng 7 2019

Bài Làm:

a, \(P=\frac{x+3}{\sqrt{x}-2}:\left(\frac{\sqrt{x}-1}{\sqrt{x}+2}+\frac{5\sqrt{x}-2}{x-4}\right)\)

\(=\frac{x+3}{\sqrt{x}-2}:\left(\frac{\sqrt{x}-1}{\sqrt{x}+2}+\frac{5\sqrt{x}-2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\right)\)

\(=\frac{x+3}{\sqrt{x}-2}:\frac{\left(\sqrt{x}-1\right)\left(\sqrt{x}-2\right)+5\sqrt{x}-2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\)

\(=\frac{x-3}{\sqrt{x}-2}:\frac{x-3\sqrt{x}+2+5\sqrt{x}-2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\)

\(=\frac{x-3}{\sqrt{x}-2}:\frac{x+2\sqrt{x}}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}\)

\(=\frac{x+3}{\sqrt{x}-2}:\frac{\sqrt{x}\left(\sqrt{x}+2\right)}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}=\frac{x+3}{\sqrt{x}-2}:\frac{\sqrt{x}}{\sqrt{x}-2}\)

\(=\frac{x+3}{\sqrt{x}-2}.\frac{\sqrt{x}-2}{\sqrt{x}}=\frac{x+3}{\sqrt{x}}\)

23 tháng 5 2021

Mình ghi nhầm. \(x=\frac{\sqrt{4+2\sqrt{3}}.\left(\sqrt{3}-1\right)}{\sqrt{6+2\sqrt{5}}-\sqrt{5}}\)nhé

5 tháng 4 2020

a) Đkxđ : \(\left\{{}\begin{matrix}a\ge0\\a\ne9\end{matrix}\right.\)

A = \(\left(\frac{\sqrt{a}+3}{\left(\sqrt{a}-3\right)\left(\sqrt{a}+3\right)}+\frac{\sqrt{a}-3}{\left(\sqrt{a}-3\right)\left(\sqrt{a}+3\right)}\right)\left(1-\frac{3}{\sqrt{a}}\right)\)

= \(\frac{2\sqrt{a}}{\left(\sqrt{a}-3\right)\left(\sqrt{a}+3\right)}.\frac{\sqrt{a}-3}{\sqrt{a}}\)

= \(\frac{2}{\sqrt{a}+3}\)

b) Để A > \(\frac{1}{2}\)

<=> \(\frac{2}{\sqrt{a}+3}>\frac{1}{2}\Leftrightarrow\frac{2}{\sqrt{a}+3}-\frac{1}{2}>0\)

<=> \(4-\sqrt{a}-3>0\Leftrightarrow1-\sqrt{a}>0\Leftrightarrow a< 1\)

Vậy để A >1/2 thì a <1

20 tháng 10 2023

a: \(P=\dfrac{\sqrt{x}-\left(\sqrt{x}-1\right)}{\sqrt{x}\left(\sqrt{x}-1\right)}:\dfrac{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)-\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}\)

\(=\dfrac{1}{\sqrt{x}\left(\sqrt{x}-1\right)}\cdot\dfrac{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}{x-1-x+4}\)

\(=\dfrac{1}{\sqrt{x}}\cdot\dfrac{\sqrt{x}-2}{3}=\dfrac{\sqrt{x}-2}{3\sqrt{x}}\)

b: P=1/4

=>\(\dfrac{\sqrt{x}-2}{3\sqrt{x}}=\dfrac{1}{4}\)

=>\(4\left(\sqrt{x}-2\right)=3\sqrt{x}\)

=>\(4\sqrt{x}-8-3\sqrt{x}=0\)

=>\(\sqrt{x}=8\)

=>x=64

c: Khi \(x=4+2\sqrt{3}\) thì \(P=\dfrac{\sqrt{4+2\sqrt{3}}-2}{3\cdot\sqrt{4+2\sqrt{3}}}\)

\(=\dfrac{\sqrt{3}+1-2}{3\left(\sqrt{3}+1\right)}=\dfrac{\sqrt{3}-1}{3\sqrt{3}+3}=\dfrac{2-\sqrt{3}}{3}\)

29 tháng 8 2020

Bài làm:

Ta có: 

\(P=\left(1-\frac{x-3\sqrt{x}}{x-9}\right)\div\left(\frac{\sqrt{x}-9}{2-\sqrt{x}}+\frac{\sqrt{x}-2}{3+\sqrt{x}}-\frac{9-x}{x+\sqrt{x}-6}\right)\)

\(P=\frac{x-9-x+3\sqrt{x}}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\div\left[\frac{\left(9-\sqrt{x}\right)\left(3+\sqrt{x}\right)+\left(\sqrt{x}-2\right)^2-9+x}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+3\right)}\right]\)

\(P=\frac{3\sqrt{x}-9}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\div\frac{-x+6\sqrt{x}+27+x-4\sqrt{x}+2-9+x}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+3\right)}\)

\(P=\frac{3}{\sqrt{x}+3}\div\frac{x+2\sqrt{x}+20}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+3\right)}\)

\(P=\frac{3}{\sqrt{x}+3}\cdot\frac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+3\right)}{x+2\sqrt{x}+20}\)

\(P=\frac{3\left(\sqrt{x}-2\right)}{x+2\sqrt{x}+20}=\frac{3\sqrt{x}-6}{x+2\sqrt{x}+20}\)

16 tháng 10 2020

1: Rút gọn biểu thức

a) Ta có: \(5\sqrt{\frac{1}{5}}+\frac{1}{3}\sqrt{45}+\sqrt{\left(2-\sqrt{5}\right)^2}\)

\(=5\cdot\frac{1}{\sqrt{5}}+\frac{1}{3}\cdot3\sqrt{5}+\left|2-\sqrt{5}\right|\)

\(=\sqrt{5}+\sqrt{5}+\sqrt{5}-2\)(Vì \(2< \sqrt{5}\))

\(=3\sqrt{5}-2\)

b) Ta có: \(\frac{5+\sqrt{5}}{5-\sqrt{5}}+\frac{5-\sqrt{5}}{5+\sqrt{5}}\)

\(=\frac{\left(5+\sqrt{5}\right)^2}{\left(5-\sqrt{5}\right)\left(5+\sqrt{5}\right)}+\frac{\left(5-\sqrt{5}\right)^2}{\left(5+\sqrt{5}\right)\left(5-\sqrt{5}\right)}\)

\(=\frac{30+10\sqrt{5}+30-10\sqrt{5}}{25-5}\)

\(=\frac{60}{20}=3\)

2:

Sửa đề: \(A=\left(\frac{1}{\sqrt{x}-1}-\frac{1}{\sqrt{x}}\right):\left(\frac{\sqrt{x}+1}{\sqrt{x}-2}-\frac{\sqrt{x}+2}{\sqrt{x}-1}\right)\)

a) Ta có: \(A=\left(\frac{1}{\sqrt{x}-1}-\frac{1}{\sqrt{x}}\right):\left(\frac{\sqrt{x}+1}{\sqrt{x}-2}-\frac{\sqrt{x}+2}{\sqrt{x}-1}\right)\)

\(=\left(\frac{\sqrt{x}}{\sqrt{x}\left(\sqrt{x}-1\right)}-\frac{\sqrt{x}-1}{\sqrt{x}\left(\sqrt{x}-1\right)}\right):\left(\frac{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}-\frac{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}\right)\)

\(=\frac{\sqrt{x}-\sqrt{x}+1}{\sqrt{x}\left(\sqrt{x}-1\right)}:\frac{x-1-\left(x-4\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}\)

\(=\frac{1}{\sqrt{x}\left(\sqrt{x}-1\right)}:\frac{3}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}\)

\(=\frac{1}{\sqrt{x}\left(\sqrt{x}-1\right)}\cdot\frac{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}{3}\)

\(=\frac{\sqrt{x}-2}{3\sqrt{x}}\)

b) ĐKXĐ: \(\left\{{}\begin{matrix}x>0\\x\notin\left\{1;4;\frac{14\pm6\sqrt{5}}{4}\right\}\end{matrix}\right.\)

Để \(A>\frac{1}{6}\) thì \(A-\frac{1}{6}>0\)

\(\Leftrightarrow\frac{\sqrt{x}-2}{3\sqrt{x}}-\frac{1}{6}>0\)

\(\Leftrightarrow\frac{2\sqrt{x}-4}{6\sqrt{x}}-\frac{\sqrt{x}}{6\sqrt{x}}>0\)

\(\Leftrightarrow\frac{\sqrt{x}-4}{6\sqrt{x}}>0\)

\(6\sqrt{x}>0\forall x\) thỏa mãn ĐKXĐ

nên \(\sqrt{x}-4>0\)

\(\Leftrightarrow\sqrt{x}>4\)

hay x>16

Kết hợp ĐKXĐ, ta được: x>16

Vậy: Để \(A>\frac{1}{6}\)thì x>16

12 tháng 12 2019

a)\(M=\left(\frac{\sqrt{x}+3}{\sqrt{x}-2}+\frac{\sqrt{x}+2}{3-\sqrt{x}}+\frac{\sqrt{x}+2}{x-5\sqrt{x}+6}\right):\left(1-\frac{\sqrt{x}}{\sqrt{x}+1}\right)\)

\(=\left(\frac{x-9}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-3\right)}-\frac{x-4}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-3\right)}+\frac{\sqrt{x}+2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-3\right)}\right):\left(\frac{\sqrt{x}+1-\sqrt{x}}{\sqrt{x}+1}\right)\)

\(=\frac{\sqrt{x}-3}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-3\right)}.\left(\sqrt{x}+1\right)\)

\(=\frac{\sqrt{x}+1}{\sqrt{x}-2}\)

b)\(\frac{1}{M}=\frac{\sqrt{x}-2}{\sqrt{x}+1}=\frac{\sqrt{x}+1-3}{\sqrt{x}+1}=1-\frac{3}{\sqrt{x}+1}\)

Ta có: \(\sqrt{x}\ge0,\forall x\ge0\)

\(\Leftrightarrow\sqrt{x}+1\ge1\)

\(\Leftrightarrow\frac{1}{\sqrt{x}+1}\le1\)

\(\Leftrightarrow\frac{3}{\sqrt{x}+1}\le3\)

\(\Leftrightarrow-\frac{3}{\sqrt{x}+1}\ge-3\)

\(\Leftrightarrow1-\frac{3}{\sqrt{x}+1}\ge-2\)

Dấu "=" xảy ra khi x=0

Vậy \(Min_{\frac{1}{M}}=-2\) khi x=0

13 tháng 12 2019

Thankssss!!