\(A=2x^2+y^2-2xy+x+2\)

\(A=\left(x^2-2xy+y^2\right)+\left[x^2+2.x.\frac{1}{2}+\left(\frac{1}{2}\right)^2\right]+\frac{7}{4}\)

\(A=\left(x-y\right)^2+\left(x+\frac{1}{2}\right)^2+\frac{7}{4}\)

Ta có: \(\hept{\begin{cases}\left(x-y\right)^2\ge0\forall x;y\\\left(x+\frac{1}{2}\right)^2\ge0\forall x\end{cases}}\Rightarrow\left(x-y\right)^2+\left(x+\frac{1}{2}\right)^2+\frac{7}{4}=A\ge\frac{7}{4}>0\forall x;y\)

Vậy không có các số tự nhiên thỏa mã đẳng thức \(A=2x^2+y^2-2xy+x+2=0\)

\(P=\frac{1}{x}+\frac{1}{y}+xy^2+x^2y=\left(\frac{1}{16x}+xy^2\right)+\left(\frac{1}{16y}+x^2y\right)+\frac{15}{16}\left(\frac{1}{x}+\frac{1}{y}\right)\)

\(\ge\frac{y}{2}+\frac{x}{2}+\frac{15}{16}.\frac{4}{x+y}\)

\(=\left(\frac{x+y}{2}+\frac{1}{2\left(x+y\right)}\right)+\frac{13}{4\left(x+y\right)}\)

\(\ge1+\frac{13}{4}=\frac{17}{4}\)

Dấu "=" xảy ra <=> x = y = 1/2

x^2 + 2y^2 - 2xy + 2x + 2 - 4y =0 
<=>x^2 + y^2 - 2xy+2x-2y+y^2-2y+1+1=0 
<=>(x-y)^2+2(x-y)+1+(y-1)^2=0 
<=>(x-y+1)^2+(y-1)^2=0 
<=>y=1;x=0

\(3xy+x+15y-44=0\)

\(\Leftrightarrow x\left(3y+1\right)+5\left(3y+1\right)=49\)

\(\Leftrightarrow\left(3y+1\right)\left(x+5\right)=49\)

Vì x;y là số nguyên

\(\Rightarrow\hept{\begin{cases}3y+1\\x+5\end{cases}\in}Z\)

\(\Rightarrow\hept{\begin{cases}3y+1\\x+5\end{cases}\in}\text{Ư}\left(49\right)=\left\{\pm1;\pm7;\pm49\right\}\)

Tự lập bảng giá trị nhé

dạ ook. mơn bn nha