\(M=\dfrac{x^2-\sqrt{x}}{x+\sqrt{x}+1}-\dfrac{x^2+\sqrt{x}}{x-\sqrt{x}+1}+x+1\)

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

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

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

\(\sqrt{2x+3}+\sqrt{2x+2}=1\)

\(\Leftrightarrow\sqrt{2x+3}-1+\sqrt{2x+2}=0\)

\(\Leftrightarrow\dfrac{2x+3-1}{\sqrt{2x+3}-1}+\dfrac{2x+2}{\sqrt{2x+2}}=0\)

\(\Leftrightarrow\dfrac{2\left(x+1\right)}{\sqrt{2x+3}-1}+\dfrac{2\left(x+1\right)}{\sqrt{2x+2}}=0\)

\(\Leftrightarrow2\left(x+1\right)\left(\dfrac{1}{\sqrt{2x+3}-1}+\dfrac{1}{\sqrt{2x+2}}\right)=0\)

Pt \(\dfrac{1}{\sqrt{2x+3}-1}+\dfrac{1}{\sqrt{2x+2}}=0\) vô nghiệm

=> x + 1 = 0

<=> x = -1

\(\sqrt{x+4}-\sqrt{2x-6}=1\)

\(\Leftrightarrow\sqrt{x+4}-3-\sqrt{2x-6}+2=0\)

\(\Leftrightarrow\dfrac{x+4-9}{\sqrt{x+4}+3}-\dfrac{2x-6-4}{\sqrt{2x-6}+2}=0\)

\(\Leftrightarrow\dfrac{x-5}{\sqrt{x+4}+3}-\dfrac{2\left(x-5\right)}{\sqrt{2x-6}+2}=0\)

\(\Leftrightarrow\left(x-5\right)\left(\dfrac{1}{\sqrt{x+4}+3}-\dfrac{2}{\sqrt{2x-6}+2}\right)=0\)

Pt \(\dfrac{1}{\sqrt{x+4}+3}-\dfrac{2}{\sqrt{2x-6}+2}=0\) vô nghiệm

=> x - 5 = 0

<=> x = 5 (nhận)

Thách oOo KiRitO oOo tính kiểu đấy đấy

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

\(=1+\left[\dfrac{\sqrt{a}\left(\sqrt{a}+1\right)\left(2\sqrt{a}-1\right)}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+a+1\right)}-\dfrac{\left(\sqrt{a}+1\right)\left(2\sqrt{a}-1\right)}{\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)}\right]\)\(\times\dfrac{\sqrt{a}\left(\sqrt{a}-1\right)}{2\sqrt{a}-1}\)

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

\(=\dfrac{\sqrt{a}+a+1+a\sqrt{a}+a-a-a\sqrt{a}-\sqrt{a}}{a+\sqrt{a}+1}\)

\(=\dfrac{a+1}{a+\sqrt{a}+1}\)

(^~^)

\(\dfrac{a+1}{a+\sqrt{a}+1}=\dfrac{\sqrt{6}}{1+\sqrt{6}}\)

<=> \(a+\sqrt{6}a+1+\sqrt{6}=\sqrt{6}a+\sqrt{6a}+\sqrt{6}\)

<=> \(a-\sqrt{6a}+1=0\left(1\right)\)

<=> \(\left(\sqrt{a}-\sqrt{2+\sqrt{3}}\right)\left(\sqrt{a}-\sqrt{2-\sqrt{3}}\right)=0\)

<=> \(\left[{}\begin{matrix}a=2+\sqrt{3}\\a=2-\sqrt{3}\end{matrix}\right.\)

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

\(=\left[\dfrac{\left(\sqrt{a}\right)-\left(\sqrt{a}-1\right)}{\sqrt{a}\left(\sqrt{a}-1\right)}\right]\div\left[\dfrac{\left(a-1\right)-\left(a-2\right)}{\left(\sqrt{a}-1\right)\left(\sqrt{a}-2\right)}\right]\)

\(=\dfrac{1}{\sqrt{a}\left(\sqrt{a}-1\right)}\times\dfrac{\left(\sqrt{a}-1\right)\left(\sqrt{a}-2\right)}{1}\)

\(=\dfrac{\sqrt{a}-2}{\sqrt{a}}\)

(^~^)

\(\dfrac{\sqrt{a}-2}{\sqrt{a}}>\dfrac{1}{6}\)

\(\Leftrightarrow6\sqrt{a}-12>\sqrt{a}\)

\(\Leftrightarrow5\sqrt{a}>12\)

\(\Leftrightarrow a>\dfrac{144}{25}\)

a)

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

\(=\left[\dfrac{\left(1-\sqrt{a}\right)\left(1+\sqrt{a}+a\right)}{1-\sqrt{a}}+\sqrt{a}\right]\left[\dfrac{\left(1+\sqrt{a}\right)\left(1-\sqrt{a}+a\right)}{1+\sqrt{a}}-\sqrt{a}\right]\)

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

\(=\left(\sqrt{a}+1\right)^2\left(\sqrt{a}-1\right)^2\)

\(=\left(a-1\right)^2\)

b)

\(\left(a-1\right)^2< 7-4\sqrt{3}\)

\(\Leftrightarrow\left(a-1\right)^2< \left(2-\sqrt{3}\right)^2\)

\(\Leftrightarrow a-1< 2-\sqrt{3}\)

\(\Leftrightarrow a< 3-\sqrt{3}\)

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