Question

3x²+2y²+2xy-10x-10y+15=0

Answer 1

3x² + 2y² + 2xy - 10x - 10y + 15 = 0

<=> 6x² + 4y² + 4xy - 20x - 20y + 30 = 0

<=> (4x² + 4xy + y²) - 10(2x + y) + 25 + (5y² - 10xy + 5) = 0

<=> (2x + y)² - 10(2x + y) + 25 + 5(y - 1)² = 0

<=> (2x + y - 5)² + 5(y - 1)² = 0

<=> \(\hept{\begin{cases}2x+y-5=0\\y-1=0\end{cases}}\)

<=> \(\hept{\begin{cases}x=\frac{5-y}{2}\\y=1\end{cases}}\)

<=> \(\hept{\begin{cases}x=2\\y=1\end{cases}}\)

Answer 2

\(3x^2+2y^2+2xy-10x-10y+15=0\)

\(\Rightarrow\left(x^2+2xy+y^2-6x-6y+9\right)+\left(2x^2-4x+2\right)+\left(y^2-4y+4\right)=0\)

\(\Rightarrow\left(x+y-3\right)^2+2\left(x-1\right)^2+\left(y-2\right)^2=0\)

mà \(\left(x+y-3\right)^2\ge0\forall x,y\)

\(2\left(x-1\right)^2\ge0\forall x\)

\(\left(y-2\right)^2\ge0\forall y\)

\(\Rightarrow\hept{\begin{cases}x+y-3=0\\x-1=0\\y-2=0\end{cases}\Rightarrow\hept{\begin{cases}x=1\\y=2\end{cases}}}\)