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Bài 4. Cho biểu thức: P = ((a + 2sqrt(a))/(sqrt(a) + 2) - 1) / ((a - sqrt(a))/(sqrt(a) - 1) + 1)
Với 
a: \(P=\left(\dfrac{2+\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\right):\dfrac{\sqrt{x}+1-\sqrt{x}}{\sqrt{x}+1}\)
\(=\dfrac{1}{\sqrt{x}-1}\cdot\dfrac{\sqrt{x}+1}{1}=\dfrac{\sqrt{x}+1}{\sqrt{x}-1}\)
b: Để P nguyên thì \(\sqrt{x}+1⋮\sqrt{x}-1\)
\(\Leftrightarrow\sqrt{x}-1\in\left\{-1;1;2\right\}\)
hay \(x\in\left\{0;4;9\right\}\)
ĐKXĐ: \(x>0;x\ne9\)
\(P=\left(\dfrac{x+7}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-3\right)}-\dfrac{4\left(\sqrt{x}+1\right)}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-3\right)}+\dfrac{\sqrt{x}-3}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-3\right)}\right)\left(\dfrac{\sqrt{x}+6}{\sqrt{x}}\right)\)
\(=\left(\dfrac{x+7-4\sqrt{x}-4+\sqrt{x}-3}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-3\right)}\right)\left(\dfrac{\sqrt{x}+6}{\sqrt{x}}\right)\)
\(=\left(\dfrac{x-3\sqrt{x}}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-3\right)}\right).\left(\dfrac{\sqrt{x}+6}{\sqrt{x}}\right)\)
\(=\dfrac{\sqrt{x}\left(\sqrt{x}-3\right)}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-3\right)}.\dfrac{\left(\sqrt{x}+6\right)}{\sqrt{x}}\)
\(=\dfrac{\sqrt{x}+6}{\sqrt{x}+1}\)
b.
Ta có \(P=\dfrac{\sqrt{x}+1+5}{\sqrt{x}+1}=1+\dfrac{5}{\sqrt{x}+1}\)
Do \(\sqrt{x}+1>0\Rightarrow\dfrac{5}{\sqrt{x}+1}>0\Rightarrow P>1\)
\(P=\dfrac{6\left(\sqrt{x}+1\right)-5\sqrt{x}}{\sqrt{x}+1}=6-\dfrac{5\sqrt{x}}{\sqrt{x}+1}\)
Do \(\left\{{}\begin{matrix}5\sqrt{x}>0\\\sqrt{x}+1>0\end{matrix}\right.\) ;\(\forall x>0\Rightarrow\dfrac{5\sqrt{x}}{\sqrt{x}+1}>0\)
\(\Rightarrow P< 6\Rightarrow1< P< 6\)
Mà P nguyên \(\Rightarrow P=\left\{2;3;4;5\right\}\)
- Để \(P=2\Rightarrow\dfrac{\sqrt{x}+6}{\sqrt{x}+1}=2\Rightarrow\sqrt{x}+6=2\sqrt{x}+2\Rightarrow x=16\)
- Để \(P=3\Rightarrow\dfrac{\sqrt{x}+6}{\sqrt{x}+1}=3\Rightarrow\sqrt{x}+6=3\sqrt{x}+3\Rightarrow\sqrt{x}=\dfrac{3}{2}\Rightarrow x=\dfrac{9}{4}\)
- Để \(P=4\Rightarrow\dfrac{\sqrt{x}+6}{\sqrt{x}+1}=4\Rightarrow\sqrt{x}+6=4\sqrt{x}+4\Rightarrow\sqrt{x}=\dfrac{2}{3}\Rightarrow x=\dfrac{4}{9}\)
- Để \(P=5\Rightarrow\dfrac{\sqrt{x}+6}{\sqrt{x}+1}=5\Rightarrow\sqrt{x}+6=5\sqrt{x}+5\Rightarrow\sqrt{x}=\dfrac{1}{4}\Rightarrow x=\dfrac{1}{16}\)
a) ĐKXĐ: \(x>0,x\ne1\)
\(P=\dfrac{x-2\sqrt{x}}{x\sqrt{x}-1}+\dfrac{\sqrt{x}+1}{x\sqrt{x}+x+\sqrt{x}}+\dfrac{1+2x-2\sqrt{x}}{x^2-\sqrt{x}}\)
\(=\dfrac{x-2\sqrt{x}}{\left(\sqrt{x}\right)^3-1}+\dfrac{\sqrt{x}+1}{\sqrt{x}\left(x+\sqrt{x}+1\right)}+\dfrac{1+2x-2\sqrt{x}}{\sqrt{x}\left(\left(\sqrt{x}\right)^3-1\right)}\)
\(=\dfrac{x-2\sqrt{x}}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}+\dfrac{\sqrt{x}+1}{\sqrt{x}\left(x+\sqrt{x}+1\right)}+\dfrac{1+2x-2\sqrt{x}}{\sqrt{x}\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\)
\(=\dfrac{\left(x-2\sqrt{x}\right)\sqrt{x}+\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)+1+2x-2\sqrt{x}}{\sqrt{x}\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\)
\(=\dfrac{x\sqrt{x}+x-2\sqrt{x}}{\sqrt{x}\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}=\dfrac{\sqrt{x}\left(x+\sqrt{x}-2\right)}{\sqrt{x}\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\)
\(=\dfrac{\sqrt{x}\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)}{\sqrt{x}\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}=\dfrac{\sqrt{x}+2}{x+\sqrt{x}+1}\)
b) Ta có: \(\left\{{}\begin{matrix}\sqrt{x}+2>0\\x+\sqrt{x}+1>0\end{matrix}\right.\Rightarrow P>0\)
Vì \(x>0\Rightarrow2x+\sqrt{x}>0\Rightarrow2x+2\sqrt{x}+2-\left(\sqrt{x}+2\right)>0\)
\(\Rightarrow2\left(x+\sqrt{x}+1\right)>\sqrt{x}+2\Rightarrow\dfrac{\sqrt{x}+2}{x+\sqrt{x}+1}< 2\)
mà P nguyên \(\Rightarrow\dfrac{\sqrt{x}+2}{x+\sqrt{x}+1}=1\Rightarrow\sqrt{x}+2=x+\sqrt{x}+1\)
\(\Rightarrow x-1=0\Rightarrow x=1\) mà \(x\ne1\Rightarrow\) không có x để P nguyên
Mn ơi giúp mình với. Please!!!!!!!!!!!!!!!!!!!!!!!!!!
với a ≥ 0, a ≠ 4
a:
ĐKXĐ: \(\left\{{}\begin{matrix}a>0\\a\ne4\end{matrix}\right.\)
\(A=\left(\dfrac{\sqrt{a}}{\sqrt{a}-2}+\dfrac{\sqrt{a}}{\sqrt{a}+2}\right)\cdot\dfrac{a-4}{\sqrt{4a}}\)
\(=\dfrac{\sqrt{a}\left(\sqrt{a}+2\right)+\sqrt{a}\left(\sqrt{a}-2\right)}{\left(\sqrt{a}+2\right)\left(\sqrt{a}-2\right)}\cdot\dfrac{a-4}{2\sqrt{a}}\)
\(=\dfrac{a+2\sqrt{a}+a-2\sqrt{a}}{a-4}\cdot\dfrac{a-4}{2\sqrt{a}}=\dfrac{2a}{2\sqrt{a}}=\sqrt{a}\)
b: A-2<0
=>\(\sqrt{a}-2< 0\)
=>\(\sqrt{a}< 2\)
=>0<=a<4
kết hợp ĐKXĐ, ta được: 0<a<4
c: Để \(\dfrac{4}{A+1}=\dfrac{4}{\sqrt{a}+1}\) là số nguyên thì
\(\sqrt{a}+1\inƯ\left(4\right)\)
=>\(\sqrt{a}+1\in\left\{1;-1;2;-2;4;-4\right\}\)
=>\(\sqrt{a}\in\left\{0;-2;1;-3;3;-5\right\}\)
=>\(\sqrt{a}\in\left\{0;1;3\right\}\)
=>\(a\in\left\{0;1;9\right\}\)
Kết hợp ĐKXĐ, ta được: \(a\in\left\{1;9\right\}\)
a) \(A=\left(\dfrac{\sqrt{a}}{\sqrt{a}-2}+\dfrac{\sqrt{a}}{\sqrt{a}+2}\right)\cdot\dfrac{a-4}{\sqrt{4a}}\left(dkxd:a\ge0;a\ne4\right)\)
\(=\left[\dfrac{\sqrt{a}\left(\sqrt{a}+2\right)}{\left(\sqrt{a}-2\right)\left(\sqrt{a}+2\right)}+\dfrac{\sqrt{a}\left(\sqrt{a}-2\right)}{\left(\sqrt{a}-2\right)\left(\sqrt{a}+2\right)}\right]\cdot\dfrac{a-4}{2\sqrt{a}}\)
\(=\dfrac{a+2\sqrt{a}+a-2\sqrt{a}}{a-4}\cdot\dfrac{a-4}{2\sqrt{a}}\)
\(=\dfrac{2a}{2\sqrt{a}}\)
\(=\sqrt{a}\)
b) Để \(A-2< 0\) thì: \(\sqrt{a}-2< 0\)
\(\Rightarrow\sqrt{a}< 2\)
\(\Rightarrow a< 4\)
Kết hợp với điều kiện xác định của \(a\), ta được: \(0\le a< 4\)
c) Để \(\dfrac{4}{A+1}\) nguyên thì \(\dfrac{4}{\sqrt{a}+1}\) nguyên
\(\Rightarrow4⋮\sqrt{a}+1\)
\(\Rightarrow\sqrt{a}+1\inƯ\left(4\right)\)
Mà \(\sqrt{a}+1\ge1\forall a\ge0;a\ne4\)
\(\Rightarrow\sqrt{a}+1\in\left\{1;2;4\right\}\)
\(\Rightarrow\sqrt{a}\in\left\{0;1;3\right\}\)
\(\Rightarrow a\in\left\{0;1;9\right\}\)
Kết hợp với điều kiện xác định của \(a\), ta được: \(a\in\left\{0;1;9\right\}\)
\(\text{#}Toru\)
i don't now
mong thông cảm !
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