a. \(A=\left(\dfrac{1}{\sqrt{x}-1}-\dfrac{1}{\sqrt{x}}\right):\left(\dfrac{\sqrt{x}+1}{\sqrt{x}-2}-\dfrac{\sqrt{x}+2}{\sqrt{x}-1}\right)\) (ĐKXĐ: \(x>0;x\ne1;x\ne4\))
\(=\left[\dfrac{\sqrt{x}}{\sqrt{x}\left(\sqrt{x}-1\right)}-\dfrac{\sqrt{x}-1}{\sqrt{x}\left(\sqrt{x}-1\right)}\right]:\left[\dfrac{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}-2\right)}-\dfrac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}-2\right)}\right]\)
\(=\dfrac{\sqrt{x}-\sqrt{x}+1}{\sqrt{x}\left(\sqrt{x}-1\right)}:\dfrac{x-1-\left(x-4\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}-2\right)}\)
\(=\dfrac{1}{\sqrt{x}\left(\sqrt{x}-1\right)}:\dfrac{x-1-x+4}{\left(\sqrt{x}-1\right)\left(\sqrt{x}-2\right)}\)
\(=\dfrac{1}{\sqrt{x}\left(\sqrt{x}-1\right)}\cdot\dfrac{\left(\sqrt{x}-1\right)\left(\sqrt{x}-2\right)}{3}\)
\(=\dfrac{\sqrt{x}-2}{3\sqrt{x}}\)
Vậy \(A=\dfrac{\sqrt{x}-2}{3\sqrt{x}}\).
\(---\)
b. Ta có: \(A=0\Leftrightarrow\dfrac{\sqrt{x}-2}{3\sqrt{x}}=0\)
\(\Leftrightarrow\sqrt{x}-2=0\)
\(\Leftrightarrow\sqrt{x}=2\)
\(\Leftrightarrow x=4\left(ktm\right)\)
Vậy không thể tìm được giá trị nào của \(x\) để \(A=0\).
\(---\)
c. Ta có: \(A< 0\Leftrightarrow\dfrac{\sqrt{x}-2}{3\sqrt{x}}< 0\)
\(\Leftrightarrow\sqrt{x}-2< 0\left(vì.3\sqrt{x}>0\right)\)
\(\Leftrightarrow\sqrt{x}< 2\)
\(\Leftrightarrow x< 4\)
Kết hợp với điều kiện xác định của \(x\), ta được:
\(0< x< 4;x\ne1\)
Vậy \(A< 0\) khi \(0< x< 4;x\ne1\).
a) \(A=\left(\dfrac{1}{\sqrt{x}-1}-\dfrac{1}{\sqrt{x}}\right):\left(\dfrac{\sqrt{x}+1}{\sqrt{x}-2}-\dfrac{\sqrt{x}+2}{\sqrt{x}-1}\right)\) (ĐK: \(x>0;x\ne1;x\ne4\))
\(A=\left[\dfrac{\sqrt{x}}{\sqrt{x}\left(\sqrt{x}-1\right)}-\dfrac{\sqrt{x}-1}{\sqrt{x}\left(\sqrt{x}-1\right)}\right]:\left[\dfrac{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}-\dfrac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}\right]\)
\(A=\dfrac{\sqrt{x}-\sqrt{x}+1}{\sqrt{x}\left(\sqrt{x}-1\right)}:\dfrac{x-1-x+4}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}\)
\(A=\dfrac{1}{\sqrt{x}\left(\sqrt{x}-1\right)}\cdot\dfrac{\left(\sqrt{x}-2\right)\left(\sqrt{x}-1\right)}{3}\)
\(A=\dfrac{\sqrt{x}-2}{3\sqrt{x}}\)
b) \(A=0\) khi
\(\dfrac{\sqrt{x}-2}{3\sqrt{x}}=0\)
\(\Rightarrow\sqrt{x}-2=0\)
\(\Rightarrow\sqrt{x}=2\)
\(\Rightarrow x=4\left(ktm\right)\)
c) \(A< 0\) khi
\(\dfrac{\sqrt{x}-2}{3\sqrt{x}}< 0\)
\(\Rightarrow\sqrt{x}-2< 0\)
\(\Rightarrow\sqrt{x}< 2\)
\(\Rightarrow x< 4\)
kết hợp với đk:
\(0< x< 4,x\ne1\)
