P=(1/(a−1)+(3căna+5)/(acăna−a−căna+1)[(căna+1)^2/4căna)(a>0, a khác1

* Hướng dẫn giải

a) Với a > 0 và a ≠ 1 ta có:

$\begin{array}{l}P=\left[\frac{\sqrt{a}-1}{(a-1)(\sqrt{a}-1)}+\frac{3\sqrt{a}+5}{(a-1)(\sqrt{a}-1)}\right].\frac{(a+2\sqrt{a}+1)-4\sqrt{a}}{4\sqrt{a}}\\ =\frac{4\sqrt{a}+4}{{(\sqrt{a}-1)}^2(\sqrt{a}+1)}.\frac{a-2\sqrt{a}+1}{4\sqrt{a}}=\frac{4}{{(\sqrt{a}-1)}^2}.\frac{{(\sqrt{a}-1)}^2}{4\sqrt{a}}\\ =\frac{1}{\sqrt{a}}\end{array}$

b, Có $Q=\frac{a-\sqrt{a}+1}{\sqrt{a}}$

Xét $Q-1=\frac{a-2\sqrt{a}+1}{\sqrt{a}}=\frac{{(\sqrt{a}-1)}^2}{\sqrt{a}}$

Vì ${(\sqrt{a}-1)}^2>0,\sqrt{a}>0,\forall a>0,a\ne 1\Rightarrow Q-1>0\Rightarrow Q>1$