tex]...

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решить симметрическую систему уравнений:

[tex] \left \{ {{ x^{2} + xy + y^{2} = 3} \atop {xy( x^{2} + y^{2}) = 2 }} \right.

[/tex]

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Artem112

Verified answer

$\left \{ {{ x^{2} + xy + y^{2} = 3} \atop {xy( x^{2} + y^{2}) = 2 }} \right.
\\\
x^2+y^2=a
\\\
 xy=b
\\\
 \left \{ {{ a+b= 3} \atop {ab= 2 }} \right.
\\\
a=3-b
\\\
(3-b)b=2
\\\
b^2-3b+2=0
\\\
D=9-8=1
\\\
b_1=2
\\\
b_2=1
\\\
a_1=1
\\\
a_2=2$
$x^2+y^2=2
\\\
 xy=1
\\\
x= \frac{1}{y} 
\\\
 \frac{1}{y^2} +y^2-2=0
\\\
y^4-2y^2+1=0
\\\
(y^2-1)^2=0
\\\
y^2=1
\\\
y_1=1
\\\
y_2=-1
\\\
x_1=1
\\\
x_2=-1$
$x^2+y^2=1
\\\
 xy=2
\\\
x= \frac{2}{y} 
\\\
 \frac{4}{y^2} +y^2-2=0
\\\
4y^4-2y^2+1=0
\\\
4z^2-2z+1=0
\\\
D_1=1-4$
Ответ: (1; 1); (-1; -1)

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