a) \(A=\left(\frac{\sqrt{a}}{2}-\frac{1}{2\sqrt{a}}\right)^2\cdot\left(\frac{\sqrt{a}-1}{\sqrt{a}+1}-\frac{\sqrt{a}+1}{\sqrt{a}-1}\right)\)
\(A=\left(\frac{a-1}{2\sqrt{a}}\right)^2\cdot\left[\frac{\left(\sqrt{a}-1\right)^2-\left(\sqrt{a}+1\right)^2}{\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)}\right]\)
\(A=\frac{\left(a-1\right)^2\cdot\left(-4\sqrt{a}\right)}{4a\cdot\left(a-1\right)}\)
\(A=\frac{-\left(a-1\right)}{\sqrt{a}}=\frac{-a+1}{\sqrt{a}}\)
b) \(A< 0\Leftrightarrow\frac{-a+1}{\sqrt{a}}< 0\Leftrightarrow-a+1< 0\Leftrightarrow a>1\)
c) \(A=-2\Leftrightarrow\frac{-a+1}{\sqrt{a}}=-2\)
\(\Leftrightarrow-a+1=-2\sqrt{a}\)
\(\Leftrightarrow a-2\sqrt{a}-1=0\)
\(\Leftrightarrow\left(\sqrt{a}-1\right)^2-2=0\)
\(\Leftrightarrow\left(\sqrt{a}-1\right)^2=2\)
Vì \(\sqrt{a}-1\ge-1\Rightarrow\sqrt{a}-1=\sqrt{2}\Leftrightarrow a=3+2\sqrt{2}\) (t/m)
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