Question

tìm x,y biết x² +y²+xy-x+y+1=0

Answer 1

Nhân 2 vế của pt cho 2 : \(2x^2+2y^2+2xy-2x+2y+1=0\)

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

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

Vì \(\left(x+y\right)^2,\left(x-1\right)^2,\left(y+1\right)^2\ge0\)nên pt (1) có nghiệm khi và chỉ khi \(\hept{\begin{cases}\left(x+y\right)^2=0\\\left(x-1\right)^2=0\\\left(y+1\right)^2=0\end{cases}\Leftrightarrow\hept{\begin{cases}x=-y\\x=1\\y=-1\end{cases}}}\)

Vậy pt có nghiệm duy nhất (x;y)=(1;-1)

Answer 2

\(x^2+y^2+xy-x+y+1=0\)

\(\Leftrightarrow2x^2+2y^2+2xy-2x+2y+2=0\)

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

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

Vì \(\hept{\begin{cases}\left(x+y\right)^2\ge0\\\left(x-1\right)^2\ge0\\\left(y+1\right)^2\ge0\end{cases}}\Rightarrow\left(x+y\right)^2+\left(x-1\right)^2+\left(y+1\right)^2\ge0\forall x;y\)

Dấu "=" xảy ra \(\Leftrightarrow\hept{\begin{cases}\left(x+y\right)^2=0\\\left(x-1\right)^2=0\\\left(y+1\right)^2=0\end{cases}}\Leftrightarrow\hept{\begin{cases}x+y=0\\x=1\\y=-1\end{cases}}\left(tm\right)\)

Vậy pt có nghiệm là x = 1 ; y = - 1

Answer 3

x² + y² + xy - x + y + 1 = 0

<=> 2( x² + y² + xy - x + y + 1 ) = 2.0

<=> 2x² + 2y² + 2xy - 2x + 2y + 2 = 0

<=> ( x² + 2xy + y² ) + ( x² - 2x + 1 ) + ( y² + 2y + 1 ) = 0

<=> ( x + y )² + ( x - 1 )² + ( y + 1 )² = 0 (*)

Vì \(\hept{\begin{cases}\left(x+y\right)^2\\\left(x-1\right)^2\\\left(y+1\right)^2\end{cases}}\ge0\forall x,y\Rightarrow\left(x+y\right)^2+\left(x-1\right)^2+\left(y+1\right)^2\ge0\)

Đẳng thức xảy ra ( tức (*) ) <=> \(\hept{\begin{cases}x+y=0\\x-1=0\\y+1=0\end{cases}}\Leftrightarrow\hept{\begin{cases}x=1\\y=-1\end{cases}}\)

=> x = 1 ; y = -1