Question

Решить систему:
$\left \{ {{x^3+3x^2y=20} \atop {3xy^2+y^3=7}} \right.$

Answer 1

$+\left \{ {{x^{3}+3x^{2}y=20} \atop {3xy^{2}+y^{3}=7}} \right.\\--------\\x^{3}+3x^{2}y+3xy^{2}+y^{3}=27\\\\(x+y)^{3}=27\\\\x+y=3$

$\left \{ {{x=3-y} \atop {3*(3-y)*y^{2}+y^{3}=7}} \right. \\\\\left \{ {{x=3-y} \atop {9y{^2}-3y^{3}+y^{3} =7}} \right.\\\\\left \{ {{x=3-y} \atop {2y^{3}-9y^{2}+7=0  }} \right. \\\\2y^{3}-9y^{2}+7=0\\\\(y-1)(2y^{2}-7y-7)=0\\\\1)y-1=0\\\\y_{1} =1\\\\x_{1}=3-1=2$

$2)2y^{2}-7y-7=0\\\\D=(-7)^{2}-4*2*(-7)=49+56=105\\\\y_{2}=\frac{7+\sqrt{105}}{4} \\\\y_{3}=\frac{7-\sqrt{105}}{4}\\\\x_{2}=3-\frac{7+\sqrt{105}}{4}=\frac{5-\sqrt{105}}{4}\\\\x_{3}=3-\frac{7-\sqrt{105}}{4}=\frac{5+\sqrt{105}}{4} \\\\Otvet:\boxed {(2;1),(\frac{5-\sqrt{105}}{4};\frac{7+\sqrt{105}}{4}),(\frac{5+\sqrt{105}}{4};\frac{7-\sqrt{105}}{4})}$