Giải thích các bước giải:
a.ĐKXĐ: $a\ge 0,a\notin\{4,9\}$
Ta có:
$P=\dfrac{2\sqrt{a}-9}{a-5\sqrt{a}+6}-\dfrac{\sqrt{a}+3}{\sqrt{a}-2}-\dfrac{2\sqrt{a}+1}{3-\sqrt{a}}$
$\to P=\dfrac{2\sqrt{a}-9}{a-2\sqrt{a}-3\sqrt{a}+6}-\dfrac{(\sqrt{a}-3)(\sqrt{a}+3)}{(\sqrt{a}-3)(\sqrt{a}-2)}+\dfrac{2\sqrt{a}+1}{\sqrt{a}-3}$
$\to P=\dfrac{2\sqrt{a}-9}{(\sqrt{a}-3)(\sqrt{a}-2)}-\dfrac{a-9}{(\sqrt{a}-3)(\sqrt{a}-2)}+\dfrac{(2\sqrt{a}+1)(\sqrt{a}-2)}{(\sqrt{a}-3)(\sqrt{a}-2)}$
$\to P=\dfrac{2\sqrt{a}-9}{(\sqrt{a}-3)(\sqrt{a}-2)}-\dfrac{a-9}{(\sqrt{a}-3)(\sqrt{a}-2)}+\dfrac{2a-3\sqrt{a}-2}{(\sqrt{a}-3)(\sqrt{a}-2)}$
$\to P=\dfrac{2\sqrt{a}-9-(a-9)+2a-3\sqrt{a}-2}{(\sqrt{a}-3)(\sqrt{a}-2)}$
$\to P=\dfrac{a-\sqrt{a}-2}{(\sqrt{a}-3)(\sqrt{a}-2)}$
$\to P=\dfrac{a-2\sqrt{a}+\sqrt{a}-2}{(\sqrt{a}-3)(\sqrt{a}-2)}$
$\to P=\dfrac{(\sqrt{a}-2)(\sqrt{a}+1)}{(\sqrt{a}-3)(\sqrt{a}-2)}$
$\to P=\dfrac{\sqrt{a}+1}{\sqrt{a}-3}$
b.Để $P<1$
$\to \dfrac{\sqrt{a}+1}{\sqrt{a}-3}<1$
$\to \dfrac{\sqrt{a}+1}{\sqrt{a}-3}-1<0$
$\to \dfrac{\sqrt{a}+1-(\sqrt{a}-3)}{\sqrt{a}-3}<0$
$\to \dfrac{4}{\sqrt{a}-3}<0$
$\to \sqrt{a}-3<0$ vì $4>0$
$\to \sqrt{a}<3$
$\to 0\le a< 9$
