Question

[Lewther]

1. Cho \(Q=\left(\dfrac{2\sqrt{a}}{\sqrt{a}+3}+\dfrac{\sqrt{a}}{\sqrt{a}-3}-\dfrac{3a+3}{a-9}\right):\left(\dfrac{2\sqrt{a}-2}{\sqrt{a}-3}-1\right)\)

a) Rút gọn Q.

b) Tìm x để \(Q< \dfrac{-1}{2}\)

c) Tìm Q_min

2. \(P=\dfrac{\sqrt{x}+2}{\sqrt{x}+1}-\dfrac{\sqrt{x}+3}{5-\sqrt{x}}-\dfrac{3x+4\sqrt{x}-5}{x-4\sqrt{x}-5}\)

a) Rút gọn P.

b) Tìm x để P > -2

c) Tìm số chính phương x để P có giá trị nguyên.

Answer 1

[Trần Thiên Kim]

1.

a. \(Q=\left(\dfrac{2\sqrt{a}}{\sqrt{a}+3}+\dfrac{\sqrt{a}}{\sqrt{a}-3}-\dfrac{3a+3}{a-9}\right):\left(\dfrac{2\sqrt{a}-2}{\sqrt{a}-3}-1\right)=\left(\dfrac{2\sqrt{a}}{\sqrt{a}+3}+\dfrac{\sqrt{a}}{\sqrt{a}-3}-\dfrac{3a+3}{\left(\sqrt{a}-3\right)\left(\sqrt{a}+3\right)}\right):\left(\dfrac{2\sqrt{a}-2-\sqrt{a}+3}{\sqrt{a}-3}\right)=\)

\(\dfrac{2\sqrt{a}\left(\sqrt{a}-3\right)+\sqrt{a}\left(\sqrt{a}+3\right)-3a-3}{\left(\sqrt{a}-3\right)\left(\sqrt{a}+3\right)}:\dfrac{\sqrt{a}+1}{\sqrt{a}-3}=\dfrac{2a-6\sqrt{a}+a+3\sqrt{a}-3a-3}{\left(\sqrt{a}-3\right)\left(\sqrt{a}+3\right)}.\dfrac{\sqrt{a}-3}{\sqrt{a}+1}=\dfrac{-3\sqrt{a}-3}{\left(\sqrt{a}-3\right)\left(\sqrt{a}+1\right)}.\dfrac{\sqrt{a}-3}{\sqrt{a}+1}=\dfrac{-3\left(\sqrt{a}+1\right)}{\left(\sqrt{a}-3\right)\left(\sqrt{a}+3\right)}.\dfrac{\sqrt{a}-3}{\sqrt{a}+1}=-\dfrac{3}{\sqrt{a}+3}\)

Answer 2

[Trần Thiên Kim]

Lewther

\(\dfrac{-\sqrt{x}-2}{\sqrt{x}-5}>-2\) (ĐKXĐ: \(x\ge0\) và \(x\ne25\))

\(\Leftrightarrow\dfrac{-\sqrt{x}-2}{\sqrt{x}-5}+2>0\Leftrightarrow\dfrac{-\sqrt{x}-2+2\left(\sqrt{x}-5\right)}{\sqrt{x}-5}>0\Rightarrow-\sqrt{x}-2+2\sqrt{x}-10>0\Leftrightarrow\sqrt{x}-12>0\Leftrightarrow\sqrt{x}>12\Leftrightarrow x>144\)

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