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Dasdas123123123
@Dasdas123123123
August 2022
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решите пж
x^2y+xy^2=12
xy+x+y=7
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nelle987
Verified answer
Xy = u, x + y = v
x^2 y + xy^2 = xy(x + y) = uv
uv = 12
u + v = 7
По теореме Виета u, v - корни уравнения
t^2 - 7t + 12 = 0
D = 49 - 4 * 12 = 49 - 48 = 1
t = (7 +- 1)/2
t = 3 или t = 4.
1) x + y = 3, xy = 4
Опять, по теореме Виета x, y - корни уравнения t^2 - 3t + 4 = 0
D = 9 - 16 < 0, решений нет
2) x + y = 4, xy = 3
... t^2 - 4t + 3 = 0
t^2 - 4t + 4 = 1
(t - 2)^2 = 1
t - 2 = +-1
t = 2 +- 1
t = 1 или t = 3.
Ответ. (1, 3), (3, 1).
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Answers & Comments
Verified answer
Xy = u, x + y = vx^2 y + xy^2 = xy(x + y) = uv
uv = 12
u + v = 7
По теореме Виета u, v - корни уравнения
t^2 - 7t + 12 = 0
D = 49 - 4 * 12 = 49 - 48 = 1
t = (7 +- 1)/2
t = 3 или t = 4.
1) x + y = 3, xy = 4
Опять, по теореме Виета x, y - корни уравнения t^2 - 3t + 4 = 0
D = 9 - 16 < 0, решений нет
2) x + y = 4, xy = 3
... t^2 - 4t + 3 = 0
t^2 - 4t + 4 = 1
(t - 2)^2 = 1
t - 2 = +-1
t = 2 +- 1
t = 1 или t = 3.
Ответ. (1, 3), (3, 1).