Система..

[Гост]

| x^2+4xy+5y^2=8
| y^2-2x-4y=0

Благодаря предварително

[ammornil]

[tex]\left|x^2+4.x.y+5.y^2=8 \\ y^2-2.x-4.y=0 \right, \\
\cyr{DM:} \hspace{12} x \in R, \hspace{4} y \in R \\
\left|x^2+4.x.y+5.y^2=8 \\ y^2-2.x-4.y=0 \right, \hspace{8} \Rightarrow \hspace{8} \left|x^2+2.x.(2.y)+4.y^2+y^2=8 \\ y^2-2.(x+2.y)=0 \right, \hspace{8} \Rightarrow \hspace{8} \left|(x+2.y)^2+y^2=8 \\ y^2-2.(x+2.y)=0 \right, \\
\cyr{Polagame} \hspace{8} \left|y^2=u (>0) \\ x+2.y=v \right, \hspace{4} \cyr{i za originalnata sistema poluchavame} \\
\left|v^2+u=8 \\ u-2.v=0 \right, \hspace{8} \Rightarrow \hspace{8} \left|u=2.v \\ v^2+2.v-8=0 \right, \hspace{8} \Rightarrow \hspace{8} \left| u=2.v \\ v_{_{1,2}}=-1 \pm 3 \right, \\
\left|u=2.v \\ v=-4 \right, \hspace{8} \cup \hspace{8} \left|u=2.v \\ v=2 \right, \\
\left|u=-8 \notin Du \\ v=-4 \right, \hspace{8} \cup \hspace{8} \left|u=4 \\ v=2 \right, \\
\Rightarrow \left|y^2=4 \\ x+2.y=2 \right, \hspace{4} \Rightarrow \\
\left|y=-2 \\ x-4=2 \right, \hspace{8} \cup \hspace{8} \left|y=2 \\ x+4=2 \right, \\
\left|y=-2 \\ x=6 \right, \hspace{20} \cup \hspace{8} \left|y=2 \\ x=-2 \right,
\\***\\
\cyr{OTG:} \hspace{4} (-2;2), (6;-2)[/tex]