`#tnvt`
`\sqrt{x+1}+\sqrt{2x+3}=\sqrt{x+2}+\sqrt{2x+2}`
`\text{Điều kiện xác định:}`
`{(x+1>=0),(2x+3>=0),(x+2>=0),(2x+2>=0):}`
`<=>{(x>=-1),(x>=-3/2),(x>=-2),(x>=-1):}`
`=>ĐKXĐ: x>=-1`
`\sqrt{x+1}+\sqrt{2x+3}=\sqrt{x+2}+\sqrt{2x+2}` `(1)`
Đặt `x+1=a(a>=0)`
`2x+2=b(b>=0)`
Biểu thức `(1)` viết lại
`<=>\sqrt{a}+\sqrt{b+1}=\sqrt{a+1}+\sqrt{b}`
`<=>\sqrt{a}-\sqrt{b}=\sqrt{a+1}-\sqrt{b+1}`
`=>a-2\sqrt{ab}+b=a+1-2\sqrt{(a+1)(b+1)}+b+1`
`<=>a+b-a-b-1-1=-2\sqrt{ab+a+b+1}+2\sqrt{ab}`
`<=>-2=-2(\sqrt{ab+a+b+1}-\sqrt{ab})`
`<=>1=\sqrt{ab+a+b+1}-\sqrt{ab}`
`<=>\sqrt{ab}+1=\sqrt{ab+a+b+1}`
`=>ab+1+2\sqrt{ab}=ab+a+b+1`
`<=>2\sqrt{ab}=ab-ab+a+b+1-1`
`<=>2\sqrt{ab}=a+b`
`<=>a-2\sqrt{ab}+b=0`
`<=>(\sqrt{a}-\sqrt{b})^2=0`
`<=>\sqrt{a}-\sqrt{b}=0`
`<=>\sqrt{a}=\sqrt{b}`
`<=>\sqrt{x+1}=\sqrt{2x+2}`
`=>x+1=2x+2`
`<=>2x-x=1-2`
`<=>x=-1(tm)`
Vậy `S={-1}`
