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Bài 50:
\(\dfrac{5}{\sqrt{10}}=\dfrac{5\sqrt{10}}{10}=\dfrac{\sqrt{10}}{2}\)
\(\dfrac{5}{2\sqrt{5}}=\dfrac{\sqrt{5}}{2}\)
\(\dfrac{1}{3\sqrt{20}}=\dfrac{1}{6\sqrt{5}}=\dfrac{\sqrt{5}}{30}\)
\(\dfrac{2\sqrt{2}+2}{5\sqrt{2}}=\dfrac{\sqrt{2}\left(2+\sqrt{2}\right)}{5\sqrt{2}}=\dfrac{2+\sqrt{2}}{5}\)
\(\dfrac{1}{x^2+9x+20}+\dfrac{1}{x^2+11x+30}+\dfrac{1}{x^2+13x+42}=\dfrac{1}{18}\)
\(\dfrac{1}{\left(x+4\right)\left(x+5\right)}+\dfrac{1}{\left(x+5\right)\left(x+6\right)}+\dfrac{1}{\left(x+6\right)\left(x+7\right)}=\dfrac{1}{18}\)
\(\dfrac{1}{x+4}-\dfrac{1}{x+5}+\dfrac{1}{x+5}-\dfrac{1}{x+6}+\dfrac{1}{x+6}-\dfrac{1}{x+7}=\dfrac{1}{18}\)
\(\dfrac{1}{x+4}-\dfrac{1}{x+7}=\dfrac{1}{18}\)
\(\dfrac{18\left(x+7-x-4\right)}{18\left(x+4\right)\left(x+7\right)}=\dfrac{\left(x+4\right)\left(x+7\right)}{18\left(x+4\right)\left(x+7\right)}\)
\(18.3=\left(x+4\right)\left(x+7\right)\)
\(x^2+11x+28-54=0\)
\(x^2+11x-26=0\)
\(\left(x-2\right)\left(x+13\right)=0\)
\(\left[{}\begin{matrix}x=2\\x=-13\end{matrix}\right.\)
Theo đề x < 0 nên x = -13
\(\dfrac{2}{3}\sqrt{4x^2-20}+2\sqrt{\dfrac{x^2-5}{9}}-3\sqrt{x^2-5}=2\)
\(\Leftrightarrow\dfrac{2}{3}.2\sqrt{x^2-5}+2\dfrac{\sqrt{x^2-5}}{3}-3\sqrt{x^2-5}=2\)
\(\Leftrightarrow\dfrac{4}{3}\sqrt{x^2-5}+\dfrac{2}{3}\sqrt{x^2-5}-3\sqrt{x^2-5}=2\)
\(\Leftrightarrow\left(\dfrac{4}{3}+\dfrac{2}{3}-3\right)\sqrt{x^2-5}=2\)
\(\Leftrightarrow-\sqrt{x^2-5}=2\)
Vì \(-\sqrt{x^2-5}\) \(\le\)0 nên mình nghĩ phương trình vô \(\eta\) nhé :))
\(\dfrac{2}{3}\sqrt{4x^2-20}+2\sqrt{\dfrac{x^2-5}{9}}-3\sqrt{x^2-5}=2\)
\(\Leftrightarrow\dfrac{4}{3}\sqrt{x^2-5}+\dfrac{2}{3}\sqrt{x^2-5}-3\sqrt{x^2-5}=2\)
\(\Leftrightarrow-\sqrt{x^2-5}=2\Leftrightarrow x^2=9\Leftrightarrow x=\pm3\)
1: \(=3\left(x+\dfrac{2}{3}\sqrt{x}+\dfrac{1}{3}\right)\)
\(=3\left(x+2\cdot\sqrt{x}\cdot\dfrac{1}{3}+\dfrac{1}{9}+\dfrac{2}{9}\right)\)
\(=3\left(\sqrt{x}+\dfrac{1}{3}\right)^2+\dfrac{2}{3}>=3\cdot\dfrac{1}{9}+\dfrac{2}{3}=1\)
Dấu '=' xảy ra khi x=0
2: \(=x+3\sqrt{x}+\dfrac{9}{4}-\dfrac{21}{4}=\left(\sqrt{x}+\dfrac{3}{2}\right)^2-\dfrac{21}{4}>=-3\)
Dấu '=' xảy ra khi x=0
3: \(A=-2x-3\sqrt{x}+2< =2\)
Dấu '=' xảy ra khi x=0
5: \(=x-2\sqrt{x}+1+1=\left(\sqrt{x}-1\right)^2+1>=1\)
Dấu '=' xảy ra khi x=1
bài 1
biểu thức có nghĩa khi x, y thỏa mãn đồng thời
\(\left\{{}\begin{matrix}x,y\ne0\\\dfrac{y}{x}\ge0\end{matrix}\right.\Rightarrow x.y>0}\)x, y khác 0
x.y>0
a: \(\Leftrightarrow\dfrac{2x-3}{x-1}=4\)
=>4x-4=2x-3
=>2x=1
hay x=1/2
b: \(\Leftrightarrow\sqrt{\dfrac{2x-3}{x-1}}=2\)
=>(2x-3)=4x-4
=>4x-4=2x-3
=>2x=1
hay x=1/2(nhận)
c: \(\Leftrightarrow\sqrt{2x+3}\left(\sqrt{2x-3}-2\right)=0\)
=>2x+3=0 hoặc 2x-3=4
=>x=-3/2 hoặc x=7/2
e: \(\Leftrightarrow2\sqrt{x-5}+\sqrt{x-5}-\sqrt{x-5}=4\)
=>căn (x-5)=2
=>x-5=4
hay x=9
b) \(B=\dfrac{x-\sqrt{x}}{1-\sqrt{x}}-\dfrac{x\sqrt{x}}{\sqrt{x}}=\dfrac{\sqrt{x}\left(x-\sqrt{x}\right)-x\sqrt{x}\left(1-\sqrt{x}\right)}{\sqrt{x}\left(1-\sqrt{x}\right)}\) = \(\dfrac{x\sqrt{x}-x-x\sqrt{x}+x^2}{\sqrt{x}-x}=\dfrac{x^2-x}{\sqrt{x}-x}\)
c) \(C=\dfrac{x+2\sqrt{x}}{\sqrt{x}-x}-\dfrac{x\sqrt{x}}{\sqrt{x}+1}=\dfrac{\left(\sqrt{x}+1\right)\left(x+2\sqrt{x}\right)-x\sqrt{x}\left(\sqrt{x}-x\right)}{\left(\sqrt{x}-x\right)\left(\sqrt{x}+1\right)}=x+2\sqrt{x}-x\sqrt{x}\)
\(d,D=\dfrac{x+2\sqrt{x}}{\sqrt{x}+2}+\dfrac{5\sqrt{x}-2}{x-4}=\dfrac{x+2\sqrt{x}}{\sqrt{x}+2}+\dfrac{5\sqrt{x}-2}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}=\) \(\dfrac{\left(x+2\sqrt{x}\right)\left(\sqrt{x}-2\right)+5\sqrt{x}-2}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}=\dfrac{x+7\sqrt{x}-2}{\sqrt{x}+2}\)
e) \(E=\dfrac{\sqrt{x}}{\sqrt{x}-3}+\dfrac{\sqrt{x}-24}{x-9}=\dfrac{\sqrt{x}}{\sqrt{x}-3}+\dfrac{\sqrt{x}-24}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}=\dfrac{\sqrt{x}\left(\sqrt{x}+3\right)+\sqrt{x}-24}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\) = \(\dfrac{2\sqrt{x}-24}{\sqrt{x}+3}\)
F) F = \(\dfrac{3}{\sqrt{x}+5}+\dfrac{20-2\sqrt{x}}{x-25}=\dfrac{3\left(\sqrt{x}-5\right)+20-2\sqrt{x}}{\left(\sqrt{x}+5\right)\left(\sqrt{x}-5\right)}=\dfrac{23-2\sqrt{x}}{\sqrt{x}+5}\)
\(\sqrt{4\cdot5}-\sqrt{25\cdot\dfrac{1}{5}}-\dfrac{\sqrt{5\cdot2}}{\sqrt{2}}\)
\(=2\sqrt{5}-\sqrt{5}-\sqrt{5}\)
=0
a: Sửa đề: \(5\dfrac{1}{5}-\dfrac{1}{2}\sqrt{20}+\sqrt{5}\)
\(=5.2-\dfrac{1}{2}\cdot2\sqrt{5}+\sqrt{5}=5.2\)
b: \(=\dfrac{1}{2}\sqrt{2}+\dfrac{3}{2}\sqrt{2}+\dfrac{5}{2}\sqrt{2}=\dfrac{9}{2}\sqrt{2}\)
c: \(=2\sqrt{5}-3\sqrt{5}+9\sqrt{2}+\sqrt{77}=-\sqrt{5}+9\sqrt{2}+\sqrt{77}\)
d: \(=\dfrac{1}{10}\cdot10\sqrt{2}+\dfrac{2}{5}\sqrt{2}+0.4\cdot5\sqrt{2}\)
\(=\dfrac{17}{5}\sqrt{2}\)
Ta có: \(\dfrac{20}{x}-\dfrac{20}{x+2}=\dfrac{10}{3}\)
Suy ra: \(10x\left(x+2\right)=60\left(x+2\right)-60x\)
\(\Leftrightarrow10x^2+20x-120=0\)
\(\Leftrightarrow x^2+2x-12=0\)
\(\Leftrightarrow\left(x+1\right)^2=13\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\sqrt{13}-1\left(nhận\right)\\x=-\sqrt{13}-1\left(nhận\right)\end{matrix}\right.\)