Đáp án:

$\displaystyle \begin{array}{{>{\displaystyle}l}} VD\ 12:\\ a.P=\frac{\sqrt{x} -1}{2\sqrt{x}}\\ b.\ P=\frac{1}{3}\\ VD\ 13:\\ a.P=\frac{-\sqrt{x}}{\sqrt{x} +\sqrt{y}}\\ b.\ P=-\frac{2}{5}\\ VD\ 14:\\ a.P=\frac{2-\sqrt{x}}{\sqrt{x} +2}\\ b.\ x=36\ \end{array}$

Giải thích các bước giải:

$\displaystyle \begin{array}{{>{\displaystyle}l}} VD\ 12:\\ a.\ ĐKXĐ:\ x\geqslant 0;\ x\neq 1\\ P=\frac{\left(\sqrt{x} +2\right)\left(\sqrt{x} -1\right) -\left(\sqrt{x} -2\right)\left(\sqrt{x} +1\right)}{\left(\sqrt{x} +1\right)\left(\sqrt{x} -1\right)^{2}} .\frac{( x-1)^{2}}{4x}\\ P=\frac{x+\sqrt{x} -2-x+\sqrt{x} +2}{\left(\sqrt{x} +1\right)\left(\sqrt{x} -1\right)^{2}} .\frac{( x-1)^{2}}{4x}\\ P=\frac{2\sqrt{x}( x-1)}{4x\left(\sqrt{x} -1\right)} =\frac{\sqrt{x} -1}{2\sqrt{x}}\\ b.\ |x-5|=4\\ TH\ 1:\ x-5=4\Leftrightarrow x=9\Rightarrow P=\frac{3-1}{2.3} =\frac{1}{3}\\ TH\ 2:\ \ x-5=-4\Leftrightarrow x=1\ ( loại)\\ VD\ 13:\\ a.\ ĐKXĐ:\ x\geqslant 0;\ y\geqslant 0;\ x\neq y\\ P=\frac{4\sqrt{xy} -\left(\sqrt{x} +\sqrt{y}\right)^{2}}{2\left(\sqrt{x} +\sqrt{y}\right)\left(\sqrt{x} -\sqrt{y}\right)} .\frac{2\sqrt{x}}{\sqrt{x} -\sqrt{y}}\\ P=\frac{2\sqrt{xy} -x-y}{\left(\sqrt{x} +\sqrt{y}\right)\left(\sqrt{x} -\sqrt{y}\right)} .\frac{\sqrt{x}}{\sqrt{x} -\sqrt{y}}\\ P=\frac{-\left(\sqrt{x} -\sqrt{y}\right)^{2}}{\left(\sqrt{x} +\sqrt{y}\right)\left(\sqrt{x} -\sqrt{y}\right)} .\frac{\sqrt{x}}{\sqrt{x} -\sqrt{y}}\\ P=\frac{-\sqrt{x}}{\sqrt{x} +\sqrt{y}}\\ b.\ \frac{1}{P} =\frac{\sqrt{x} +\sqrt{y}}{-\sqrt{x}} =-1-\sqrt{\frac{y}{x}} =-1-\frac{3}{2} =-\frac{5}{2}\\ \Rightarrow P=-\frac{2}{5}\\ VD\ 14:\\ a.\ x\geqslant 0;\ x\neq 4\\ P=\frac{\sqrt{x} +2-2}{\left(\sqrt{x} +2\right)^{2}} .\frac{\left(\sqrt{x} -2\right)\left(\sqrt{x} +2\right)}{2-\sqrt{x} -2}\\ P=\frac{\sqrt{x}}{\left(\sqrt{x} +2\right)^{2}} .\frac{\left(\sqrt{x} -2\right)\left(\sqrt{x} +2\right)}{-\sqrt{x}}\\ P=\frac{2-\sqrt{x}}{\sqrt{x} +2}\\ b.\ P=-\frac{1}{2} \Leftrightarrow \frac{2-\sqrt{x}}{\sqrt{x} +2} =-\frac{1}{2}\\ \Leftrightarrow 2\left(\sqrt{x} -2\right) =\sqrt{x} +2\\ \Leftrightarrow \sqrt{x} =6\\ \Leftrightarrow x=36\ ( TM) \end{array}$