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đk: x khác 0
A = \(\sqrt{\dfrac{x^4-6x^2+9+12x^2}{x^2}}+\sqrt{x^2+4x+4-8x}\)
= \(\sqrt{\dfrac{x^4+6x^2+9}{x^2}}+\sqrt{x^2-4x+4}\)
= \(\sqrt{\dfrac{\left(x^2+3\right)^2}{x^2}}+\sqrt{\left(x-2\right)^2}\)
= \(\dfrac{x^2+3}{\left|x\right|}+\left|x-2\right|\)
TH1: x \(\ge2\)
A = \(\dfrac{x^2+3}{x}+x-2\)
= \(\dfrac{x^2+3+x^2-2x}{x}=\dfrac{2x^2-2x+3}{x}\)
TH2: \(0< x< 2\)
A = \(\dfrac{x^2+3}{x}-x+2\)
= \(\dfrac{x^2+3-x^2+2x}{x}=\dfrac{2x+3}{x}\)
TH3: x < 0
A = \(\dfrac{x^2+3}{-x}-x+2\)
= \(\dfrac{-x^2-3}{x}-x+2=\dfrac{-x^2-3-x^2+2x}{x}=\dfrac{-2x^2+2x-3}{x}\)
a) A = \(\sqrt{\frac{\left(x^2-3\right)^2+12x^2}{x^2}}+\sqrt{\left(x+2\right)^2-8x}=\sqrt{\frac{\left(x^2+3\right)^2}{x^2}}+\sqrt{\left(x-2\right)^2}\)
\(=\frac{x^2+3}{\left|x\right|}+\left|x-2\right|=\left|x\right|+\frac{3}{\left|x\right|}+ \left|x-2\right|\)
b) A nhận gt nguyên khi |x| thuộc Ư(3) (các ước dương)
=> |x| thuộc {1;3} => x thuộc {-3;-1;1;3}
Điều kiện: x khác 0
\(=\sqrt{\frac{x^4-6x^2+9+12x^2}{x^2}}+\sqrt{x^2+4x+4-8x}\)
\(=\sqrt{\frac{x^4+6x^2+9}{x^2}}+\sqrt{x^2-4x+4}=\sqrt{\frac{\left(x^2+3\right)^2}{x^2}}+\sqrt{\left(x-2\right)^2}\)
\(=\left|\frac{x^2+3}{x}\right|+\left|x-2\right|=\frac{x^2+3}{\left|x\right|}+\left|x-2\right|\)
\(\sqrt{\frac{\left(x^2-3\right)^2+12x^2}{x^2}}+\sqrt{\left(x+2\right)^2-8x}\)
=\(\frac{\sqrt{x^4-6x+9+12x^2}}{\sqrt{x^2}}+\sqrt{x^2+4x+4-8x}\)
=\(\frac{\sqrt{x^4+6x+9}}{x}+\sqrt{x^2-4x+4}\)
=\(\frac{\sqrt{\left(x^2+3\right)^2}}{x}+\sqrt{\left(x-2\right)^2}\)
=\(\frac{\sqrt{\left(x^2+3\right)^2}}{x}+\left|x-2\right|\)
=\(\frac{x^2+3}{x}+\left|x-2\right|\)
TH1: x\(\ge\)2 =>|x-2|=x-2
=>\(\frac{x^2+3}{x}+\left|x-2\right|\)
=\(\frac{x^2+3}{x}+x-2\)
=\(\frac{x^2+3}{x}+\frac{x^2-2x}{x}=\frac{2x^2-2x+3}{x}\)
TH2:x\(\le\)2 =>|x-2|=2-x
=>\(\frac{x^2+3}{x}+\left|x-2\right|\)
=\(\frac{x^2+3}{x}+2-x\)
=\(\frac{x^2+3}{x}+\frac{2x-x^2}{x}=\frac{2x+3}{x}\)
`(\sqrt(3x^2-12x+12)-x+2)/(x-2)`
`=(\sqrt(3(x^2-4x+4))-(x-2))/(x-2)`
`=(\sqrt(3(x-2)^2)) -(x-2))/(x-2)`
`=(\sqrt3. (x-2) - (x-2))/(x-2)`
`=( (\sqrt3-1) (x-2))/(x-2)`
`=\sqrt3-1`
`=>` C.