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Bài 1:
a: ĐKXĐ: 2x+3>=0 và x-3>0
=>x>3
b: ĐKXĐ:(2x+3)/(x-3)>=0
=>x>3 hoặc x<-3/2
c: ĐKXĐ: x+2<0
hay x<-2
d: ĐKXĐ: -x>=0 và x+3<>0
=>x<=0 và x<>-3
1/ Rút gọn: \(a)3\sqrt{2a}-\sqrt{18a^3}+4\sqrt{\dfrac{a}{2}}-\dfrac{1}{4}\sqrt{128a}\left(a\ge0\right)=3\sqrt{2a}-3a\sqrt{2a}+2\sqrt{2a}-2\sqrt{2a}=3\sqrt{2a}\left(1-a\right)\)b)\(\dfrac{\sqrt{2}-1}{\sqrt{2}+2}-\dfrac{2}{2+\sqrt{2}}+\dfrac{\sqrt{2}+1}{\sqrt{2}}=\dfrac{\sqrt{2}-1-2}{\sqrt{2}+2}+\dfrac{\sqrt{2}+1}{\sqrt{2}}=\dfrac{\sqrt{2}-3}{\sqrt{2}+2}+\dfrac{\sqrt{2}+1}{\sqrt{2}}=\dfrac{\sqrt{2}-3+2+1+2\sqrt{2}}{\sqrt{2}\left(1+\sqrt{2}\right)}=\dfrac{3\sqrt{2}}{\sqrt{2}\left(1+\sqrt{2}\right)}=\dfrac{3}{1+\sqrt{2}}\)c)\(\dfrac{2+\sqrt{5}}{\sqrt{2}+\sqrt{3+\sqrt{5}}}+\dfrac{2-\sqrt{5}}{\sqrt{2}-\sqrt{3-\sqrt{5}}}=\dfrac{\sqrt{2}\left(2+\sqrt{5}\right)}{\left(\sqrt{2}+\sqrt{3+\sqrt{5}}\right)\sqrt{2}}+\dfrac{\sqrt{2}\left(2-\sqrt{5}\right)}{\sqrt{2}\left(\sqrt{2}-\sqrt{3-\sqrt{5}}\right)}=\dfrac{2\sqrt{2}+\sqrt{10}}{2+\sqrt{6+2\sqrt{5}}}+\dfrac{2\sqrt{2}-\sqrt{10}}{2-\sqrt{6-2\sqrt{5}}}=\dfrac{2\sqrt{2}+\sqrt{10}}{2+\sqrt{\left(\sqrt{5}+1\right)^2}}+\dfrac{2\sqrt{2}-\sqrt{10}}{2-\sqrt{\left(\sqrt{5}-1\right)^2}}=\dfrac{\sqrt{2}\left(2+\sqrt{5}\right)}{2+\sqrt{5}+1}+\dfrac{\sqrt{2}\left(2-\sqrt{5}\right)}{2-\sqrt{5}+1}=\dfrac{\sqrt{2}\left(2+\sqrt{5}\right)}{3+\sqrt{5}}+\dfrac{\sqrt{2}\left(2-\sqrt{5}\right)}{3-\sqrt{5}}=\dfrac{\sqrt{2}\left(2+\sqrt{5}\right)\left(3-\sqrt{5}\right)+\sqrt{2}\left(2-\sqrt{5}\right)\left(3+\sqrt{5}\right)}{\left(3+\sqrt{5}\right)\left(3-\sqrt{5}\right)}=\dfrac{\sqrt{2}\left(6-2\sqrt{5}+3\sqrt{5}-5+6+2\sqrt{5}-3\sqrt{5}-5\right)}{9-5}=\dfrac{2\sqrt{2}}{4}=\dfrac{1}{\sqrt{2}}\)
Làm nốt nè :3
\(2.a.P=\left(\dfrac{1}{x-\sqrt{x}}+\dfrac{1}{\sqrt{x}-1}\right):\dfrac{\sqrt{x}}{x-2\sqrt{x}+1}=\dfrac{\sqrt{x}+1}{\sqrt{x}\left(\sqrt{x}-1\right)}.\dfrac{\left(\sqrt{x}-1\right)^2}{\sqrt{x}}=\dfrac{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}{x}=\dfrac{x-1}{x}\left(x>0;x\ne1\right)\)\(b.P>\dfrac{1}{2}\Leftrightarrow\dfrac{x-1}{x}-\dfrac{1}{2}>0\)
\(\Leftrightarrow\dfrac{x-2}{2x}>0\)
\(\Leftrightarrow x-2>0\left(do:x>0\right)\)
\(\Leftrightarrow x>2\)
\(3.a.A=\left(\dfrac{\sqrt{a}}{\sqrt{a}-1}-\dfrac{\sqrt{a}}{a-\sqrt{a}}\right):\dfrac{\sqrt{a}+1}{a-1}=\dfrac{\sqrt{a}-1}{\sqrt{a}-1}.\dfrac{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}{\sqrt{a}+1}=\sqrt{a}-1\left(a>0;a\ne1\right)\)
\(b.Để:A< 0\Leftrightarrow\sqrt{a}-1< 0\Leftrightarrow a< 1\)
Kết hợp với DKXĐ : \(0< a< 1\)
Bài 4:
a: ĐKXĐ: x>=0; x<>1
b: \(P=\dfrac{2a^2+4}{1-a^3}-\dfrac{1}{\sqrt{a}+1}+\dfrac{1}{\sqrt{a}-1}\)
\(=\dfrac{2a^2+4}{-\left(a-1\right)\left(a^2+a+1\right)}+\dfrac{-\sqrt{a}+1+\sqrt{a}+1}{a-1}\)
\(=\dfrac{-2a^2-4}{\left(a-1\right)\left(a^2+a+1\right)}+\dfrac{2}{a-1}\)
\(=\dfrac{-2a^2-4+2a^2+2a+2}{\left(a-1\right)\left(a^2+a+1\right)}=\dfrac{2a+2}{\left(a-1\right)\left(a^2+a+1\right)}\)
\(1,\\ A=1+\left[\dfrac{\left(2\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}{\left(1-\sqrt{a}\right)\left(1+\sqrt{a}\right)}-\dfrac{\sqrt{a}\left(2\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}{\left(1-\sqrt{a}\right)\left(a+\sqrt{a}+1\right)}\right]\cdot\dfrac{\sqrt{a}\left(\sqrt{a}-1\right)}{2\sqrt{a}-1}\\ A=1+\left[\dfrac{2\sqrt{a}-1}{1-\sqrt{a}}-\dfrac{\sqrt{a}\left(2\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}{\left(1-\sqrt{a}\right)\left(a+\sqrt{a}+1\right)}\right]\cdot\dfrac{\sqrt{a}\left(\sqrt{a}-1\right)}{2\sqrt{a}-1}\\ A=1+\dfrac{\left(2\sqrt{a}-1\right)\left(a+\sqrt{a}+1\right)-\left(2\sqrt{a}-1\right)\left(a+\sqrt{a}\right)}{\left(1-\sqrt{a}\right)\left(a+\sqrt{a}+1\right)}\cdot\dfrac{\sqrt{a}\left(\sqrt{a}-1\right)}{2\sqrt{a}-1}\)
\(A=1+\dfrac{\left(2\sqrt{a}-1\right)\left(a+\sqrt{a}+1-a-\sqrt{a}\right)}{-\left(\sqrt{a}-1\right)\left(a+\sqrt{a}+1\right)}\cdot\dfrac{\sqrt{a}\left(\sqrt{a}-1\right)}{2\sqrt{a}-1}\\ A=1+\dfrac{-\sqrt{a}\left(2\sqrt{a}-1\right)}{\left(a+\sqrt{a}+1\right)\left(2\sqrt{a}-1\right)}\\ A=1-\dfrac{\sqrt{a}}{a+\sqrt{a}+1}=\dfrac{a+\sqrt{a}+1-\sqrt{a}}{a+\sqrt{a}+1}=\dfrac{a+1}{a+\sqrt{a}+1}\)
1 , ĐKXĐ : \(x\ge0,x\ne1\)
Với điều kiện xác định trên phương trình đã cho thánh :
\(\dfrac{1}{\sqrt{x}+1}-\dfrac{2}{\sqrt{x}-1}+\dfrac{x+3}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\)
\(=\dfrac{\sqrt{x}-1-2\left(\sqrt{x}+1\right)+x+3}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\)
\(=\dfrac{x+\sqrt{x}}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}=\dfrac{\sqrt{x}\left(\sqrt{x}+1\right)}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\)
\(=\dfrac{\sqrt{x}}{\sqrt{x}-1}\)
a) ĐKXĐ: \(x>0;x\ne1;x\ne4\), rút gọn: \(Y=\left(\sqrt{x}-3+\dfrac{1}{\sqrt{x}-1}\right):\left(\sqrt{x}-1+\dfrac{1}{1-\sqrt{x}}\right)=\dfrac{\left(\sqrt{x}-3\right)\left(\sqrt{x}-1\right)+1}{\sqrt{x}-1}:\dfrac{\left(\sqrt{x}-1\right)^2-1}{\sqrt{x}-1}=\dfrac{x-\sqrt{x}-3\sqrt{x}+3+1}{\sqrt{x}-1}.\dfrac{\sqrt{x}-1}{x-2\sqrt{x}+1-1}=\dfrac{x-4\sqrt{x}+4}{x-2\sqrt{x}}=\dfrac{\left(\sqrt{x}-2\right)^2}{\sqrt{x}\left(\sqrt{x}-2\right)}=\dfrac{\sqrt{x}-2}{\sqrt{x}}\)
\(a.Y=\left(\sqrt{x}-3+\dfrac{1}{\sqrt{x}-1}\right):\left(\sqrt{x}-1+\dfrac{1}{1-\sqrt{x}}\right)=\dfrac{x-4\sqrt{x}+3+1}{\sqrt{x}-1}:\dfrac{x-2\sqrt{x}+1-1}{\sqrt{x}-1}=\dfrac{\left(\sqrt{x}-2\right)^2}{\sqrt{x}-1}.\dfrac{\sqrt{x}-1}{\sqrt{x}\left(\sqrt{x}-2\right)}=\dfrac{\sqrt{x}-2}{\sqrt{x}}\left(x>0;x\ne1;x\ne4\right)\)
\(b.Y< \dfrac{1}{2}\Leftrightarrow\dfrac{\sqrt{x}-2}{\sqrt{x}}-\dfrac{1}{2}< 0\)
\(\Leftrightarrow\dfrac{\sqrt{x}-4}{2\sqrt{x}}< 0\)
Do : \(2\sqrt{x}>0\)\(\Leftrightarrow\sqrt{x}-4< 0\Leftrightarrow x< 16\)
Kết hợp với ĐKXĐ : \(0< x< 16\left(x\ne1;x\ne4\right)\)