Question

Cho \(P=\left(\dfrac{x}{x+2}-\dfrac{x^3-8}{x^3+8}.\dfrac{x^2-2x+4}{x^2-4}\right):\dfrac{4}{x+2}\)

a ) Rút gọn P 

b ) Tìm x để P<0

c ) Tìm x để \(P=\dfrac{1}{x}+1\)

d ) Tính P  khi \(\left|2x-1\right|=3\)

e ) Tính giá trị nhỏ nhất của P

Answer 1

`a)P=(x/(x+2)-(x^3-8)/(x^3+8)*(x^2-2x+4)/(x^2-4)):4/(x+2)`

`đk:x ne 0,x ne -2`

`P=(x/(x+2)-((x-2)(x^2+2x+4))/((x+2)(x^2-2x+4))*(x^2-2x+4)/((x-2)(x+2)))*(x+2)/4`

`=(x/(x+2)-(x^2+2x+4)/(x+2)^2)*(x+2)/4`

`=(x^2+2x-x^2-2x-4)/(x+2)^2*(x+2)/4`

`=-4/(x+2)^2*(x+2)/4`

`=-1/(x+2)`

`b)P<0`

`<=>-1/(x+2)<0`

Vì `-1<0`

`<=>x+2>0`

`<=>x> -2`

`c)P=1/x+1(x ne 0)`

`<=>-1/(x+2)=1/x+1`

`<=>1/x+1+1/(x+2)=0``

`<=>x+2+x(x+2)+x=0`

`<=>x^2+4x+2=0`

`<=>` \(\left[ \begin{array}{l}x=\sqrt2-2\\x=-\sqrt2-2\end{array} \right.\) 

`d)|2x-1|=3`

`<=>` \(\left[ \begin{array}{l}2x=4\\2x=-2\end{array} \right.\) 

`<=>` \(\left[ \begin{array}{l}x=2(l)\\x=-1(tm)\end{array} \right.\) 

`x=-1=>P=-1/(-1+2)=-1`

`e)P=-1/(x+2)` thì nhỏ nhất cái gì nhỉ?

Answer 2

a) đk: \(x\ne-2;2\)

 \(P=\left[\dfrac{x}{x+2}-\dfrac{\left(x-2\right)\left(x^2+2x+4\right)}{\left(x+2\right)\left(x^2-2x+4\right)}.\dfrac{x^2-2x+4}{\left(x-2\right)\left(x+2\right)}\right]:\dfrac{4}{x+2}\)

= \(\left[\dfrac{x}{x+2}-\dfrac{x^2+2x+4}{\left(x+2\right)^2}\right].\dfrac{x+2}{4}\)

= \(\dfrac{x^2+2x-x^2-2x-4}{\left(x+2\right)^2}.\dfrac{x+2}{4}\) = \(\dfrac{-4}{4\left(x+2\right)}=\dfrac{-1}{x+2}\)

b) Để P < 0

<=> \(\dfrac{-1}{x+2}< 0\)

<=> x +2 > 0

<=> x > -2 ( x khác 2)

c) Để P= \(\dfrac{1}{x}+1\)

<=> \(\dfrac{-1}{x+2}=\dfrac{1}{x}+1\)

<=> \(\dfrac{1}{x}+\dfrac{1}{x+2}+1=0\)

<=> \(\dfrac{x+2+x+x\left(x+2\right)}{x\left(x+2\right)}=0\)

<=> x² + 4x + 2 = 0

<=> (x+2)² = 2

<=> \(\left[{}\begin{matrix}x=\sqrt{2}-2\left(c\right)\\x=-\sqrt{2}-2\left(c\right)\end{matrix}\right.\)

d) Để \(\left|2x-1\right|=3\)

<=> \(\left[{}\begin{matrix}2x-1=3< =>x=2\left(l\right)\\2x-1=-3< =>x=-1\left(c\right)\end{matrix}\right.\)

Thay x = -1, ta có:

P = \(\dfrac{-1}{-1+2}=-1\)

 

Answer 3

a) ĐKXĐ: \(x\ne2;-2\)

\(P=\left(\dfrac{x}{x+2}-\dfrac{x^3-8}{x^3+8}.\dfrac{x^2-2x+4}{x^2-4}\right):\dfrac{4}{x+2}\)

\(=\left(\dfrac{x}{x+2}-\dfrac{\left(x-2\right)\left(x^2+2x+4\right)}{\left(x+2\right)\left(x^2-2x+4\right)}.\dfrac{x^2-2x+4}{\left(x-2\right)\left(x+2\right)}\right):\dfrac{4}{x+2}\)

\(=\left(\dfrac{x}{x+2}-\dfrac{x^2+2x+4}{x+2}.\dfrac{1}{x+2}\right):\dfrac{4}{x+2}\)

\(=\left(\dfrac{x}{x+2}-\dfrac{x^2+2x+4}{\left(x+2\right)^2}\right):\dfrac{4}{x+2}\)

\(=\dfrac{x\left(x+2\right)-x^2-2x-4}{\left(x+2\right)^2}.\dfrac{x+2}{4}=-\dfrac{4}{\left(x+2\right)^2}.\dfrac{x+2}{4}=-\dfrac{1}{x+2}\)

b) \(P< 0\Rightarrow-\dfrac{1}{x+2}< 0\Rightarrow x+2>0\Rightarrow x>-2\)

\(\Rightarrow x>-2;x\ne2\)

c) \(P=\dfrac{1}{x}+1\Rightarrow\dfrac{-1}{x+2}=\dfrac{x+1}{x}\Rightarrow-x=\left(x+2\right)\left(x+1\right)\)

\(\Rightarrow-x=x^2+3x+2\Rightarrow x^2+4x+2=0\)

\(\Delta=4^2-2.4=8\Rightarrow\left[{}\begin{matrix}x=\dfrac{-b-\sqrt{\Delta}}{2a}=\dfrac{-4-2\sqrt{2}}{2}=-2-\sqrt{2}\\x=\dfrac{-b+\sqrt{\Delta}}{2a}=\dfrac{-4+2\sqrt{2}}{2}=-2+\sqrt{2}\end{matrix}\right.\)

d) \(\left|2x-1\right|=3\Rightarrow\left[{}\begin{matrix}2x-1=3\\1-2x=3\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}x=2\\x=-1\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}P=-\dfrac{1}{2+2}=-\dfrac{1}{4}\\P=-\dfrac{1}{-1+2}=-1\end{matrix}\right.\)