Question

Giải các phương trình sau:

a) \(\dfrac{1}{x^2-2x+2}\) + \(\dfrac{2}{x^2-2x+3}\) = \(\dfrac{6}{x^2-2x+4}\)

b) \(\dfrac{x^2+2x+7}{\left(x+1\right)^2+2}\) = x² + 2x + 4

Answer 1

b) Đặt \(x^2+2x+3=a\)(a>0)

Ta có: \(\dfrac{x^2+2x+7}{\left(x+1\right)^2+2}=x^2+2x+4\)

\(\Leftrightarrow\dfrac{x^2+2x+7}{x^2+2x+1+2}=x^2+2x+4\)

\(\Leftrightarrow\dfrac{x^2+2x+7}{x^2+2x+3}=x^2+2x+4\)

\(\Leftrightarrow\dfrac{a+4}{a}=a+1\)

\(\Leftrightarrow a^2+a=a+4\)

\(\Leftrightarrow a^2=4\)

\(\Leftrightarrow\left[{}\begin{matrix}a=2\left(nhận\right)\\a=-2\left(loại\right)\end{matrix}\right.\)

\(\Leftrightarrow x^2+2x+3=2\)

\(\Leftrightarrow x^2+2x+1=0\)

\(\Leftrightarrow\left(x+1\right)^2=0\)

\(\Leftrightarrow x+1=0\)

hay x=-1

Vậy: S={-1}

Answer 2

ĐKXĐ của cả 2 pt trên đều là `x in RR`

`a,1/(x^2-2x+2)+2/(x^2-2x+3)=6/(x^2-2x+4)`

Đặt `a=x^+2x+3(a>=2)` ta có:

`1/(a-1)+2/a=6/(a+1)`

`<=>a(a+1)+2(a-1)(a+1)=6a(a-1)`

`<=>a^2+a+2(a^2-1)=6a^2-6a`

`<=>a^2+a+2a^2-2=6a^2-6a`

`<=>3a^2-5a+2=0`

`<=>3a^2-3a-2a+2=0`

`<=>3a(a-1)-2(a-1)=0`

`<=>(a-1)(3a-2)=0`

`a>=2=>a-1>=1>0`

`a>=2=>3a-2>=4>0`

Vậy pt vô nghiệm

`(x^2+2x+7)/((x+1)^2+2)=x^2+2x+4`

`<=>(x^2+2x+7)=(x^2+2x+4)(x^2+2x+3)`

Đặt `a=x^2+2x+3(a>=2)`

`pt<=>a+4=a(a+1)`

`<=>a^2+a=a+4`

`<=>a^2=4`

`<=>a=2` do `a>=2`

`<=>(x+1)^2+2=2`

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

`<=>x=-1`

Vậy `S={-1}`