b, \(A=\dfrac{\sqrt{x}}{x-4}+\dfrac{1}{\sqrt{x}-2}\)
\(=\dfrac{\sqrt{x}}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}+\dfrac{\sqrt{x}+2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\)
\(=\dfrac{2\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\)
Khi đó:
\(P=\dfrac{B}{A}=\dfrac{\dfrac{2}{\sqrt{x}-2}}{\dfrac{2\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}}=\dfrac{2}{\sqrt{x}-2}.\dfrac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}{2\left(\sqrt{x}+1\right)}=\dfrac{\sqrt{x}+2}{\sqrt{x}+1}\)
c, \(P=\dfrac{\sqrt{x}+2}{\sqrt{x}+1}=\dfrac{4}{3}\)
\(\Leftrightarrow3\sqrt{x}+6=4\sqrt{x}+4\)
\(\Leftrightarrow\sqrt{x}=2\)
\(\Leftrightarrow x=4\left(l\right)\)
Vậy không tồn tại giá trị x thỏa mãn.
d, \(\left(\sqrt{x}+1\right)P-\sqrt{x}-4\sqrt{x-1}+26=-6x+10\sqrt{5x}\)
\(\Leftrightarrow\left(\sqrt{x}+1\right).\dfrac{\sqrt{x}+2}{\sqrt{x}+1}-\sqrt{x}-4\sqrt{x-1}+26=-6x+10\sqrt{5x}\)
\(\Leftrightarrow\sqrt{x}+2-\sqrt{x}-4\sqrt{x-1}+26=-6x+10\sqrt{5x}\)
\(\Leftrightarrow6x-10\sqrt{5x}-4\sqrt{x-1}+28=0\)
\(\Leftrightarrow5x-10\sqrt{5x}+25+x-1-4\sqrt{x-1}+4=0\)
\(\Leftrightarrow\left(\sqrt{5x}-5\right)^2+\left(\sqrt{x-1}-2\right)^2=0\)
\(\Leftrightarrow\left\{{}\begin{matrix}\sqrt{5x}-5=0\\\sqrt{x-1}-2=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\sqrt{5x}=5\\\sqrt{x-1}=2\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}5x=25\\x-1=4\end{matrix}\right.\)
\(\Leftrightarrow x=5\)
a, \(x=7-4\sqrt{3}=\left(\sqrt{3}-2\right)^2\)
Khi đó: \(B=\dfrac{2}{\sqrt{x}-2}=\dfrac{2}{\sqrt{\left(\sqrt{3}-2\right)^2}-2}=\dfrac{2}{2-\sqrt{3}-2}=-\dfrac{2\sqrt{3}}{3}\)
