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a: \(P=\dfrac{\sqrt{a}\left(1-a\right)^2}{1+a}:\left[\left(a+2\sqrt{a}+1\right)\left(a-2\sqrt{a}+1\right)\right]\)
\(=\dfrac{\sqrt{a}\left(a-1\right)^2}{a+1}\cdot\dfrac{1}{\left(a-1\right)^2}=\dfrac{\sqrt{a}}{a+1}\)
b: \(P-\dfrac{1}{2}=\dfrac{\sqrt{a}}{a+1}-\dfrac{1}{2}=\dfrac{2\sqrt{a}-a-1}{2\left(a+1\right)}=\dfrac{-\left(\sqrt{a}-1\right)^2}{2\left(a+1\right)}\)
\(\Leftrightarrow a\left(P-\dfrac{1}{2}\right)< 0\)
Sửa đề; \(A=\left(\dfrac{a\sqrt{a}-3}{\left(\sqrt{a}+1\right)\left(\sqrt{a}-3\right)}-\dfrac{2\left(\sqrt{a}-3\right)}{\sqrt{a}+1}-\dfrac{\sqrt{a}+3}{\sqrt{a}-3}\right):\dfrac{a+8}{a-1}\)
\(A=\dfrac{a\sqrt{a}-3-2a+12\sqrt{a}-18-a-4\sqrt{a}-3}{\left(\sqrt{a}+1\right)\left(\sqrt{a}-3\right)}:\dfrac{a+8}{a-1}\)
\(=\dfrac{a\sqrt{a}-3a+8\sqrt{a}-24}{\left(\sqrt{a}+1\right)\left(\sqrt{a}-3\right)}\cdot\dfrac{a-1}{a+8}\)
\(=\sqrt{a}-1\)
a) \(P=\left(a+\sqrt{a}+1+\sqrt{a}\right)\left(a-\sqrt{a}+1-\sqrt{a}\right)\)
\(=\left(\sqrt{a}+1\right)^2\left(\sqrt{a}-1\right)^2=\left(a-1\right)^2\)
b) đk: a khác 1
\(P< 7-4\sqrt{3}\Leftrightarrow\left(a-1\right)^2< 7-4\sqrt{3}\Leftrightarrow a-1< \sqrt{7-4\sqrt{3}}\)
\(\Leftrightarrow a-1< \sqrt{\left(4-2\cdot2\cdot\sqrt{3}+3\right)}\Leftrightarrow a-1< 2-\sqrt{3}\Leftrightarrow a< 2-\sqrt{3}+1\)
\(\Leftrightarrow a< 3-\sqrt{3}\)
Kl:..
a: Sửa đề: \(P=\left(\dfrac{2a+1}{a\sqrt{a}+1}-\dfrac{\sqrt{a}}{a-\sqrt{a}+1}\right)\cdot\left(\dfrac{1+\sqrt{a^3}}{1+\sqrt{a}}-\sqrt{a}\right)\)
\(=\dfrac{2a+1-a-\sqrt{a}}{\left(\sqrt{a}+1\right)\left(a-\sqrt{a}+1\right)}\cdot\left(1-2\sqrt{a}+a\right)\)
\(=\dfrac{\left(\sqrt{a}-1\right)^2}{\sqrt{a}+1}\)
b: \(P\cdot\sqrt{1-a}=\dfrac{\left(\sqrt{a}-1\right)^2}{\sqrt{a}+1}\cdot\sqrt{1-a}>0\)