√х+√у=4
х+у-3√ху=1
x, y >= 0
√х = a >=0
√у = b >=0
a + b = 4
a² + b² - 3ab = a² + 2ab + b² -2ab - 3ab = 1
(a + b)² - 5ab = 1
16 - 5ab = 1
Решаем второе
5ab = 15
ab = 3
a = 4 - b
второе в первое и до конца решаем
(4 - b)b = 3
b² - 4b + 3 = 0
D = 16 - 12 = 4
b₁₂ = (4 +- 2)/2 = 1 3
1.b₁ = 1
√y = 1 y = 1
a₁ = 4 - b₁ = 3
√x = 3 x = 9
2.b₂ = 3
√y = 3 y = 9
a₂ = 4 - b₂ = 1
√x = 1 x = 1
ответ (1, 9) (9,1)
√х+√у=4
х+у-3√ху=1
x, y >= 0
√х = a >=0
√у = b >=0
a + b = 4
a² + b² - 3ab = a² + 2ab + b² -2ab - 3ab = 1
a + b = 4
(a + b)² - 5ab = 1
a + b = 4
16 - 5ab = 1
Решаем второе
5ab = 15
ab = 3
a = 4 - b
второе в первое и до конца решаем
(4 - b)b = 3
b² - 4b + 3 = 0
D = 16 - 12 = 4
b₁₂ = (4 +- 2)/2 = 1 3
1.b₁ = 1
√y = 1 y = 1
a₁ = 4 - b₁ = 3
√x = 3 x = 9
2.b₂ = 3
√y = 3 y = 9
a₂ = 4 - b₂ = 1
√x = 1 x = 1
ответ (1, 9) (9,1)