Dạng 3 Bài 1)
`B=(sqrtx/(x-4)+1/(sqrtx-2)):(sqrtx+2)/(x-4)(x>=0,x ne 4)`
`=(sqrtx/(x-4)+(sqrtx+2)/(x-4)):1/(sqrtx-2)`
`=(2sqrtx+2)/(x-4)*(sqrtx-2)`
`=(2sqrtx+2)/(sqrtx+2)`
`b)C=A(B-2)=(sqrtx+2)/(sqrtx-2)*(2sqrtx+2-2sqrtx-4)/(sqrtx+2)`
`=-2/(sqrtx-2)`
Vì `x in ZZ=>sqrtx-2 in ZZ`
`=>-2 vdots sqrtx-2`
`=>sqrtx-2 in Ư(-2)={+-1,+-2}`
`=>sqrtx in {1,3,0,4}`
`=>x in {1,9,0,16}`
Bài 2:
a) Ta có: \(B=\dfrac{\sqrt{x}+3}{\sqrt{x}+1}+\dfrac{5}{\sqrt{x}-1}+\dfrac{4}{x-1}\)
\(=\dfrac{\left(\sqrt{x}+3\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}+\dfrac{5\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}+\dfrac{4}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)
\(=\dfrac{x-\sqrt{x}+3\sqrt{x}-3+5\sqrt{x}+5+4}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)
\(=\dfrac{x+7\sqrt{x}+6}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)
\(=\dfrac{\left(\sqrt{x}+1\right)\left(\sqrt{x}+6\right)}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)
\(=\dfrac{\sqrt{x}+6}{\sqrt{x}-1}\)
b) Ta có: \(C=\left(AB+\dfrac{x-5}{\sqrt{x}-5}\right)\cdot\dfrac{\sqrt{x}-5}{\sqrt{x}}\)
\(=\left(\dfrac{\sqrt{x}-1}{\sqrt{x}-5}\cdot\dfrac{\sqrt{x}+6}{\sqrt{x}-1}+\dfrac{x-5}{\sqrt{x}-5}\right)\cdot\dfrac{\sqrt{x}-5}{\sqrt{x}}\)
\(=\dfrac{x-5+\sqrt{x}+6}{\sqrt{x}-5}\cdot\dfrac{\sqrt{x}-5}{\sqrt{x}}\)
\(=\dfrac{x+\sqrt{x}+1}{\sqrt{x}}\)
Ta có: \(C-3=\dfrac{x+\sqrt{x}+1}{\sqrt{x}}-\dfrac{3\sqrt{x}}{\sqrt{x}}\)
\(=\dfrac{x-2\sqrt{x}+1}{\sqrt{x}}\)
\(=\dfrac{\left(\sqrt{x}-1\right)^2}{\sqrt{x}}>0\forall x\) thỏa mãn ĐKXĐ
\(\Leftrightarrow C>3\)
Bài 3:
a) Ta có: \(M=\dfrac{2\sqrt{x}-9}{x-5\sqrt{x}+6}-\dfrac{\sqrt{x}+3}{\sqrt{x}-2}-\dfrac{2\sqrt{x}+1}{3-\sqrt{x}}\)
\(=\dfrac{2\sqrt{x}-9}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-3\right)}-\dfrac{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-3\right)}+\dfrac{\left(2\sqrt{x}+1\right)\left(\sqrt{x}-2\right)}{\left(\sqrt{x}-3\right)\left(\sqrt{x}-2\right)}\)
\(=\dfrac{2\sqrt{x}-9-\left(x-9\right)+2x-4\sqrt{x}+\sqrt{x}-2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-3\right)}\)
\(=\dfrac{2\sqrt{x}-9-x+9+2x-3\sqrt{x}-2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-3\right)}\)
\(=\dfrac{x-\sqrt{x}-2}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-3\right)}\)
\(=\dfrac{\left(\sqrt{x}-2\right)\left(\sqrt{x}+1\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}-3\right)}\)
\(=\dfrac{\sqrt{x}+1}{\sqrt{x}-3}\)