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\(\dfrac{x+1}{2020}+\dfrac{x+2}{2019}+\dfrac{x+3}{2018}+\dfrac{x+4}{2017}+4=0\)

⇔ \(\dfrac{x+1}{2020}+1+\dfrac{x+2}{2019}+1+\dfrac{x+3}{2018}+1+\dfrac{x+4}{2017}+1=0\)

\(\Leftrightarrow\) \(\dfrac{x+2021}{2020}+\dfrac{x+2021}{2019}+\dfrac{x+2021}{2018}+\dfrac{x+2021}{2017}=0\)

⇔ \(\left(x+2021\right)\left(\dfrac{1}{2020}+\dfrac{1}{2019}+\dfrac{1}{2018}+\dfrac{1}{2017}\right)=0\)

\(Do\) \(\left(\dfrac{1}{2020}+\dfrac{1}{2019}+\dfrac{1}{2018}+\dfrac{1}{2017}\right)\ne0\)

⇒ \(x+2021=0\)

⇔ \(x=-2021\)

\(Vậy\) \(x=-2021\)

\(\dfrac{x+1}{2021}+\dfrac{x+2}{2020}=\dfrac{x+3}{2019}+\dfrac{x+4}{2018}\)

=>\(\dfrac{x+1}{2021}+1+\dfrac{x+2}{2020}+1=\dfrac{x+3}{2019}+1+\dfrac{x+4}{2018}+1\)

=>\(\dfrac{x+2022}{2021}+\dfrac{x+2022}{2020}=\dfrac{x+2022}{2019}+\dfrac{x+2022}{2018}\)

=> (x+2022)(\(\dfrac{1}{2021}+\dfrac{1}{2020}-\dfrac{1}{2019}-\dfrac{1}{2018}\))=0

=>x+2022=0

=> x=-2022

`<=>(x+1)/2021+1+(x+2)/2020+1+(x+3)/2019+1+(x+2028)/2-3=0`

`<=>(x+2022)/2021+(x+2022)/2020+(x+2022)/2019+(x+2022)/2=0`

`<=>(x+2022)(1/2021+1/2020+1/2019+1/2)=0`

`<=>x+2022=0`

`<=>x=-2022`

<=>(x+1)/2021+1+(x+2)/2020+1+(x+3)/2019+1+(x+2028)/2-3=0

<=>(x+2022)/2021+(x+2022)/2020+(x+2022)/2019+(x+2022)/2=0

<=>(x+2022)(1/2021+1/2020+1/2019+1/2)=0

<=>x+2022=0

<=>x=-2022

d, \(\frac{x+1}{9}+\frac{x+2}{8}=\frac{x+3}{7}+\frac{x+4}{6}\)

\(\Leftrightarrow\frac{x+1}{9}+1+\frac{x+2}{8}+1=\frac{x+3}{7}+1+\frac{x+4}{6}+1\)

\(\Leftrightarrow\frac{x+10}{9}+\frac{x+10}{8}-\frac{x+10}{7}-\frac{x+10}{6}=0\)

\(\Leftrightarrow\left(x+10\right)\left(\frac{1}{9}+\frac{1}{8}-\frac{1}{7}-\frac{1}{6}\right)=0\)

\(\Leftrightarrow x+10=0\) (Vì \(\frac{1}{9}+\frac{1}{8}-\frac{1}{7}-\frac{1}{6}\) ≠ 0)

\(\Leftrightarrow x=-10\)

Vậy x = -10 là nghiệm của phương trình.

\(1,\\ b,\Leftrightarrow\left(x^2+4x+4\right)+\left(y-1\right)^2=25\\ \Leftrightarrow\left(x+2\right)^2+\left(y-1\right)^2=25\)

Vậy pt vô nghiệm do 25 ko phải tổng 2 số chính phương

\(2,\\ a,\Leftrightarrow x^2-\left(y^2-6y+9\right)=47\\ \Leftrightarrow x^2-\left(y-3\right)^2=47\)

Mà 47 ko phải hiệu 2 số chính phương nên pt vô nghiệm

\(b,\Leftrightarrow\left(x-2\right)^2+\left(3y-1\right)^2=16\)

Mà 16 ko phải tổng 2 số chính phương nên pt vô nghiệm

1a. Đề lỗi

1b. 

PT $\Leftrightarrow (x+2)^2+(y-1)^2=25$

$\Leftrightarrow (x+2)^2=25-(y-1)^2\leq 25$

$(x+2)^2$ là scp không vượt quá $25$ nên có thể nhận các giá trị $0,1,4,9,16,25$

Nếu $(x+2)^2=0\Rightarrow (y-1)^2=25$

$\Rightarrow (x,y)=(-2, 6), (-2, -4)$
Nếu $(x+2)^2=1\Rightarrow (y-1)^2=24$ không là scp (loại)

Nếu $(x+2)^2=4\Rightarrow (y-1)^2=21$ không là scp (loại)

Nếu $(x+2)^2=9\Rightarrow (y-1)^2=16$

$\Rightarrow (x,y)=(1, 5), (1, -3), (-5,5), (-5, -3)$

Nếu $(x+2)^2=25\Rightarrow (y-1)^2=0$

$\Rightarrow (x,y)=(3, 1), (-7, 1)$

Ta có: VT = \(\dfrac{x+1}{2021}\)+1 - (\(\dfrac{x+2}{2020}\)+1) = \(\dfrac{x+3}{2019}\)+1=VP
=>\(\dfrac{x+2022}{2021}+\dfrac{x+2022}{2020}-\dfrac{x+2022}{2019}=0\)
=>\(\left(x+2022\right)\left(\dfrac{1}{2021}+\dfrac{1}{2020}-\dfrac{1}{2019}\right)=0\)
=>x +2022 = 0=> x =-2022