Question

Решить систему уравнений

Answer 1

Обозначим x + y = m , xy = n

$left { {{m+n=-15} atop {m*n=-54}} right.\\left { {{m=-n-15} atop {(-n-15)*n=-54}} right.left { {{m=-n-15} atop {-n^{2}-15n+54=0}} right.\\left { {{m=-n-15} atop {n^{2}+15n-54=0}} right.\\n_{1}}=3;m_{1}=-3-15=-18\\n_{2}=-18;m_{2}=18-15=3\\\left { {{x+y=-18} atop {xy=3}} right. \\left { {{x=-y-18} atop {(-y-18)y=3}} right.\\left { {{x=-y-18} atop {-y^{2}-18y=3}} right.\\left { {{x=-y-18} atop {y^{2}+18y+3=0}} right.[tex]</p><p>[tex] y_{1}=frac{-18+sqrt{53}}{2}=-9+sqrt{53}\\x_{1}=-(-9+sqrt{53})-18=9-sqrt{53} -18=-9-sqrt{53}\\y_{2} =frac{-18-2sqrt{53}}{2}=-9-sqrt{53}\\x_{2} =-(-9-sqrt{53})-18=9+sqrt{53}-18=-9+sqrt{53} \\\left { {{x+y=3} atop {xy=-18}} right.\\left { {{x=3-y} atop {(3-y)y=-18}} right. \\left { {{x=3-y} atop {y^{2}-3y-18=0}} right. \\y_{3}=6;x_{3}=3-6=-3\\ y_{4}=-3;x_{4}=3-(-3)=3+3=6$

Ответ: $(-9-sqrt{53};-9+sqrt{53} ),(-9+sqrt{53};-9-sqrt{53}),(-3;6),(6;-3)$