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a: \(A=\dfrac{2x+2+x+\sqrt{x}+1-x+\sqrt{x}-1}{\sqrt{x}}\)
\(=\dfrac{2x+2\sqrt{x}+2}{\sqrt{x}}\)
b: \(A-5=\dfrac{2x-4\sqrt{x}+2}{\sqrt{x}}=\dfrac{2\left(\sqrt{x}-1\right)^2}{\sqrt{x}}>=0\)
=>A>=5
a: Thay \(x=\dfrac{1}{4}\) vào A, ta được:
\(A=\left(\dfrac{1}{2}+1\right):\left(\dfrac{1}{2}-2\right)=\dfrac{3}{2}:\dfrac{-3}{2}=-1\)
b: Ta có: \(B=\dfrac{\sqrt{x}+2}{\sqrt{x}-3}+\dfrac{\sqrt{x}-8}{x-5\sqrt{x}+6}\)
\(=\dfrac{x-4+\sqrt{x}-8}{\left(\sqrt{x}-3\right)\left(\sqrt{x}-2\right)}\)
\(=\dfrac{x+\sqrt{x}-12}{\left(\sqrt{x}-3\right)\left(\sqrt{x}-2\right)}\)
\(=\dfrac{\sqrt{x}+4}{\sqrt{x}-2}\)
c: Để B là số tự nhiên thì \(\sqrt{x}+4⋮\sqrt{x}-2\)
\(\Leftrightarrow\sqrt{x}-2\in\left\{1;2;3;6\right\}\)
\(\Leftrightarrow\sqrt{x}\in\left\{3;4;5;8\right\}\)
hay \(x\in\left\{16;25;64\right\}\)
a) \(Q=\) \(\left(\dfrac{\sqrt{x}+2}{x+2\sqrt{x}+1}-\dfrac{\sqrt{x}-2}{x-1}\right).\dfrac{\sqrt{x}+1}{\sqrt{x}}\left(x>0;x\ne1\right)\)
\(Q=\left(\dfrac{\sqrt{x}+2}{\left(\sqrt{x}+1\right)^2}-\dfrac{\sqrt{x}-2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\right).\dfrac{\sqrt{x}+1}{\sqrt{x}}\)
\(Q=\dfrac{\left(\sqrt{x}+2\right)\left(\sqrt{x}-1\right)-\left(\sqrt{x}-2\right)\left(\sqrt{x}+1\right)}{\left(\sqrt{x}+1\right)^2\left(\sqrt{x}-1\right)}.\dfrac{\sqrt{x}+1}{\sqrt{x}}\)
\(Q=\dfrac{x+\sqrt{x}-2-x+\sqrt{x}+2}{\left(\sqrt{x}+1\right)^2\left(\sqrt{x}-1\right)}.\dfrac{\sqrt{x}+1}{\sqrt{x}}\)
\(Q=\dfrac{2\sqrt{x}}{\left(\sqrt{x}+1\right)^2\left(\sqrt{x}-1\right)}.\dfrac{\sqrt{x}+1}{\sqrt{x}}\)
\(Q=\dfrac{2}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\) \(=\dfrac{2}{x-1}\) \(\left(đpcm\right)\).
b) Để \(Q\in Z\) <=> \(\dfrac{2}{x-1}\in Z\) <=> \(x-1\inƯ\left(2\right)=\left\{1;-1;2;-2\right\}\)
Ta có bảng sau:
x -1 | 1 | -1 | 2 | -2 |
x | 2(TM) | 0(ko TM) | 3(TM) | -1(koTM) |
Vậy để biểu thức Q nhận giá trị nguyên thì \(x\in\left\{2;3\right\}\)
\(P=A\cdot B\)
\(=\dfrac{2\sqrt{x}}{\sqrt{x}-3}\cdot\dfrac{2\sqrt{x}+6+x-3\sqrt{x}+3-5\sqrt{x}}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\)
\(=\dfrac{2\sqrt{x}}{\left(\sqrt{x}-3\right)}\cdot\dfrac{x-6\sqrt{x}+9}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}\)
\(=\dfrac{2\sqrt{x}}{\left(\sqrt{x}+3\right)}\cdot\dfrac{\left(\sqrt{x}-3\right)^2}{\left(\sqrt{x}-3\right)^2}=\dfrac{2\sqrt{x}}{\sqrt{x}+3}\)
Để P nguyên thì
\(2\sqrt{x}⋮\sqrt{x}+3\)
\(\Leftrightarrow2\sqrt{x}+6-6⋮\sqrt{x}+3\)
=>\(\sqrt{x}+3\inƯ\left(-6\right)\)
=>\(\sqrt{x}+3\in\left\{3;6\right\}\)
=>\(\sqrt{x}\in\left\{0;3\right\}\)
=>\(x\in\left\{0;9\right\}\)
Kết hợp ĐKXĐ, ta được: x=0
1: Thay x=16 vào A, ta được:
\(A=\dfrac{6-2\cdot4}{4-5}=\dfrac{-2}{-1}=2\)
(a) Với \(x\ge0,x\ne4\), ta có:
\(A=\dfrac{2x-3\sqrt{x}-2}{\sqrt{x}-2}=\dfrac{\left(\sqrt{x}-2\right)\left(2\sqrt{x}+1\right)}{\sqrt{x}-2}=2\sqrt{x}+1\)
Để \(A\le5\Rightarrow2\sqrt{x}+1\le5\)
\(\Leftrightarrow2\sqrt{x}\le4\Leftrightarrow\sqrt{x}\le2\Leftrightarrow0\le x\le4\).
Kết hợp với điều kiện thì: \(0\le x< 4.\)
(b) \(\dfrac{A}{2}=\dfrac{2\sqrt{x}+1}{2}\) nguyên khi \(\left(2\sqrt{x}+1\right)\in B\left(2\right)=\left\{0;2;4;...;2n\right\}\left(n\in N\right)\)
\(\Leftrightarrow\sqrt{x}\in\left\{-\dfrac{1}{2};\dfrac{1}{2};\dfrac{3}{2};...;\dfrac{2n+1}{2}\right\}\left(n\in N\right)\)
Hay: \(\sqrt{x}\in\left\{\dfrac{1}{2};\dfrac{3}{2};...;\dfrac{2n+1}{2}\right\}\)
\(\Leftrightarrow x\in\left\{\dfrac{1}{4};\dfrac{9}{4};...;\dfrac{\left(2n+1\right)^2}{4}\right\}\)
Để A là số nguyên thì \(3\sqrt{x}+8⋮\sqrt{x}+2\)
=>\(3\sqrt{x}+6+2⋮\sqrt{x}+2\)
=>\(2⋮\sqrt{x}+2\)
mà \(\sqrt{x}+2>2\forall x>0\)
nên A không thể là số nguyên