2x^2 + xy - y^2 =20 |*13
x^2 - 4xy + 7y^2 = 13 |*-20
x, y ≠ 0
26x^2 + 13xy - 13y^2 =260
-20x^2 + 80xy - 140y^2 = -260
Складываем
6x^2 + 93xy - 153y² = 0 | :y²
6(x/y)² + 93x/y - 153 = 0
x/y = t
6t² + 93t - 153 = 0
D = 93² + 4*6*153 = 12321 = 111²
t12 = (-93 +- 111)/12 = -17 3/2
1. x/y = -17
x = -17y
2x^2 + xy - y^2 =20
2(-17y)^2 + y*(-17y) - y^2 = 20
560y^2 = 20
y^2 = 1/28
y1 = √1/28 = √7/14
x = -17y
x1 = -17√7/14
y2 = -√7/14
x2 = 17√7/14
2. x/y = 3/2
2x^2 + xy - y^2 =20
2(3/2y)^2 + y*3/2y - y^2 = 20
9/2y^2 + 3/2y^2 - y^2 = 20
5y^2 = 20
y^2 = 4
y3 = 2
x3 = 3
y4 = -2
x4 = -3
ответ (3,2)(-3,-2)( -17√7/14, √7/14) (17√7/14, -√7/14)
a² - 5a + 4 = 0
a² - 4a - a + 4 = 0
a(a - 4) - (a - 4) = 0
(a - 1)(a - 4) = 0
a = 1; a = 4
x + y = 1; x + y = 4
b² - b - 2 = 0
b² + b - 2b - 2 = 0
b(b + 1) - 2(b + 1) = 0
(b - 2)(b + 1) = 0
b = -1; b = 2.
x - y = -1; x - y = 2
Получаем систему четырёх совокупностей:
1)
x + y = 1
x - y = -1
2x = 0
x + y = 1
x = 0
y = 1
2)
x + y = 1
x - y = 2
2x = 3
x + y = 1
x = 1,5
y = -0,5
3)
x + y = 4
x - y = -1
2x = 3
x + y = 4
x = 1,5
y = 2,5
4)
x + y = 4
x - y = 2
2x = 6
x + y = 4
x = 3
y = 1
Все системы решены алгебраическим сложением
ответ: (0; 1), (1,5; -0,5), (1,5; 2,5), (3; 1).