Ta co :(x+y)^2=(x-1)(y-1)

X^2+2xy+y^2=xy-x-y+1

2x^2+2xy+2y^2+x+y-2=0

(x^2+2xy+y^2)+(x^2+2x+1)+(y^2+2y+1)=4

(x+y)^2+(x+1)^2+(y+1)^2=4

Do x;y€Z nen (x+y)^2;(x+1)^2;(y+1)^2 la cac so chinh phuong

Suy ra co 3 truong hop

°(x+y)^2=0;(x+1)^2=0;(y+1)^2=4

°(x+y)^2=0;(x+1)^2=4;(y+1)^2=0

°(x+y)^2=4;(x+1)^2=0;(y+1)^2=0

Sau do tu giai ra tim x;y

\(PT\Leftrightarrow xy\left(x+y-1\right)+\left(x+y-1\right)=1\)

\(\Leftrightarrow\left(x+y-1\right)\left(xy+1\right)=1\)

\(\Leftrightarrow\hept{\begin{cases}x+y-1=1\\xy+1=1\end{cases}hoac\hept{\begin{cases}x+y-1=-1\\xy+1=-1\end{cases}}}\)

\(\Leftrightarrow\hept{\begin{cases}x+y=2\\xy=0\end{cases}hoac\hept{\begin{cases}x+y=0\\xy=-2\end{cases}}}\)

Đến đây thì đơn giản rồi nhé :)))

Phương trình tương đương: \(\left(x+y\right)\left(x^2y^2+1\right)=xy+2\)

\(\Leftrightarrow x+y=\frac{xu+2}{x^2y^2+1}\)

\(\Rightarrow\left(xy+2\right)⋮\left(x^2y^2+1\right)\Rightarrow\left(x^2y^2-4\right)⋮\left(x^2y^2+1\right)\)

\(\Rightarrow\left(x^2y^2+1-5\right)⋮\left(x^2y^2+1\right)\Rightarrow5⋮\left(x^2y^2+1\right)\)

\(\Rightarrow x^2y^2+1\in\left\{1;5\right\}\Rightarrow x^2y^2\in\left\{0;4\right\}\Rightarrow xy\in\left\{-2;0;2\right\}\)

Vậy: \(\left(x,y\right)\in\left\{\left(0;2\right);\left(2;0\right)\right\}\)

đenta rồi cho đenta chính phương

đenta là gì vậy bạn? Bạn có thể giải cụ thể giúp mình được ko

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↔

↔→

Nếu x+y+xy=-6→(x+1)(y+1)=-5(vì x,yϵ z nên x+1,y+1ϵ z)

ta có bảng:

x+1                   1                5                -1                  -5

y+1                 -5                -1                5                     1

x                       0                 4                 -2                    -6

y                     -6                  -2                 4                  0

→(x,y)ϵ

\(\left(1+x^2\right)\left(1+y^2+4xy\right)+2\left(x+y\right)\left(1+xy\right)=25\)

\(\Leftrightarrow\) \(x^2+2xy+y^2+x^2y^2+2xy.1+1+2\left(x+y\right)\left(1+xy\right)-25=0\)

\(\Leftrightarrow\) \(\left(x+y\right)^2+2\left(x+y\right)\left(1+xy\right)+\left(1+xy\right)^2-25=0\)

\(\Leftrightarrow\) \(\left(x+y+1+xy+5\right)\left(x+y+1+xy-5\right)=0\) \(\Rightarrow\) \(\left\{{}\begin{matrix}x+y+xy=-6\\x+y+xy=4\end{matrix}\right.\)

nếu \(x+y+xy=-6\Rightarrow\left(x+1\right)\left(y+1\right)=-5\) 

                                                                ( vì \(x,y\in Z\) nên \(x+1;y+1\in Z\) )

ta lập bảng :

\(\Rightarrow\) \(x;y\in\left\{\left(0,6\right);\left(4,-2\right);\left(-2,4\right);\left(-6,0\right)\right\}\)

Δ=(2m-2)^2-4(m-3)

=4m^2-8m+4-4m+12

=4m^2-12m+16

=4m^2-12m+9+7=(2m-3)^2+7>=7>0 với mọi m

=>Phương trình luôn có hai nghiệm phân biệt

\(\left(\dfrac{1}{x1}-\dfrac{1}{x2}\right)^2=\dfrac{\sqrt{11}}{2}\)

=>\(\dfrac{1}{x_1^2}+\dfrac{1}{x_2^2}-\dfrac{2}{x_1x_2}=\dfrac{\sqrt{11}}{2}\)

=>\(\dfrac{\left(\left(x_1+x_2\right)^2-2x_1x_2\right)}{\left(x_1\cdot x_2\right)^2}-\dfrac{2}{x_1\cdot x_2}=\dfrac{\sqrt{11}}{2}\)

=>\(\dfrac{\left(2m-2\right)^2-2\left(m-3\right)}{\left(-m+3\right)^2}-\dfrac{2}{-m+3}=\dfrac{\sqrt{11}}{2}\)

=>\(\dfrac{4m^2-8m+4-2m+6}{\left(m-3\right)^2}+\dfrac{2}{m-3}=\dfrac{\sqrt{11}}{2}\)

=>\(\dfrac{4m^2-10m+10+2m-6}{\left(m-3\right)^2}=\dfrac{\sqrt{11}}{2}\)

=>\(\sqrt{11}\left(m-3\right)^2=2\left(4m^2-8m+4\right)\)

=>\(\sqrt{11}\left(m-3\right)^2=2\left(2m-2\right)^2\)

=>\(\Leftrightarrow\left(\dfrac{m-3}{2m-2}\right)^2=\dfrac{2}{\sqrt{11}}\)

=>\(\left[{}\begin{matrix}\dfrac{m-3}{2m-2}=\sqrt{\dfrac{2}{\sqrt{11}}}\\\dfrac{m-3}{2m-2}=-\sqrt{\dfrac{2}{\sqrt{11}}}\end{matrix}\right.\)

mà m nguyên

nên \(m\in\varnothing\)

\(x^2-4xy+5y^2=2\left(x-y\right)\)

\(\Leftrightarrow x^2-4xy+5y^2-2x+2y=0\)

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

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

Vì x,y là số nguyên nên ta có các trường hợp: 

TH1: \(\hept{\begin{cases}x-2y-1=1\\y-1=1\end{cases}\Leftrightarrow}\hept{\begin{cases}x=6\\y=2\end{cases}}\)

TH2: \(\hept{\begin{cases}x-2y-1=-1\\y-1=-1\end{cases}\Leftrightarrow}\hept{\begin{cases}x=0\\y=0\end{cases}}\)

TH3: \(\hept{\begin{cases}x-2y-1=-1\\y-1=1\end{cases}\Leftrightarrow}\hept{\begin{cases}x=4\\y=2\end{cases}}\)

TH4: \(\hept{\begin{cases}x-2y-1=1\\y-1=-1\end{cases}\Rightarrow}\hept{\begin{cases}x=2\\y=0\end{cases}}\)

Vậy \(\left(x;y\right)\in\left\{\left(6;2\right),\left(0;0\right),\left(4;2\right),\left(2;0\right)\right\}\)

\(\)

x2−4xy+5y2=17x2−4xy+5y2=2

⇔(x−2y)2+y2=17⇔(x−2y)2+y2=2

= 2 + 1

= 1 + 2

Ta có bảng sau:

Vậy (x;y)={(9;4);(−7;−4);(7;4);(−9;−4);(6;1);(2;−1);(−2;1);(−6;−1)}