Задача 74669 решить систему уравнений ...

5f4e19cc6a894c0287cf31f4

12/10/2023 5:37:02 PM

решить систему уравнений

5f3ea7e3faf909182968ddd9

12/10/2023 5:50:23 PM

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[m]\left\{\begin {matrix}4y^2+2y=x^2+x\\x+y+18=0\end {matrix}\right.[/m]

Выделяем полные квадраты в первом уравнении:

[m]\left\{\begin {matrix}4(y^2+2\cdot \frac{1}{4}y+\frac{1}{16}-\frac{1}{16})=x^2+2x\cdot \frac{1}{2}+\frac{1}{4}-\frac{1}{4}\\x+y+18=0\end {matrix}\right.[/m]

[m]\left\{\begin {matrix}4(y+\frac{1}{4})^2-\frac{1}{4}=(x+ \frac{1}{2})^2-\frac{1}{4}\\x+y+18=0\end {matrix}\right.[/m]

[m]\left\{\begin {matrix}4(y+\frac{1}{4})^2=(x+ \frac{1}{2})^2\\x+y+18=0\end {matrix}\right.[/m]

Извлекаем корень:

[m]\left\{\begin {matrix}2|y+\frac{1}{4}|=|x+ \frac{1}{2}|\\x+y+18=0\end {matrix}\right.[/m]


[m]\left\{\begin {matrix}2(y+\frac{1}{4})=x+ \frac{1}{2}\\x+y+18=0\end {matrix}\right.[/m] или [m]\left\{\begin {matrix}2(y+\frac{1}{4})=-(x+ \frac{1}{2})\\x+y+18=0\end {matrix}\right.[/m]

[m]\left\{\begin {matrix}2y=x\\2y+y+18=0\end {matrix}\right.[/m] или [m]\left\{\begin {matrix}2y=-x- 1\\x=-y-18\end {matrix}\right.[/m]

[m]\left\{\begin {matrix}2y=x\\2y+y+18=0\end {matrix}\right.[/m] или [m]\left\{\begin {matrix}2y=-(-y-18)- 1\\x+y+18=0\end {matrix}\right.[/m]

[m]\left\{\begin {matrix}x=-12\\y=-6\end {matrix}\right.[/m] или [m]\left\{\begin {matrix}y=17\\x=-35\end {matrix}\right.[/m]