Решите систему уравнений: 2x4 + y2 = 10 x2 + 2y4 = 10

Решим систему методом сложения:

{2x^4 + y^2 = 10;

{x^2 + 2y^4 = 10; {2 (x^4 - y^4) - (x^2 - y^2) = 0;

{2 (x^4 + y^4) + (x^2 + y^2) = 20; {2 (x^2 + y^2) (x^2 - y^2) - (x^2 - y^2) = 0;

{2 (x^4 + y^4) + (x^2 + y^2) = 20; { (x^2 - y^2) (2 (x^2 + y^2) - 1) = 0;

{2 (x^4 + y^4) + (x^2 + y^2) = 20.

1) x^2 - y^2 = 0;

y^2 = x^2; y = ±x; 4x^4 + 2x^2 - 20 = 0; 2x^4 + x^2 - 10 = 0; D = 1^2 + 4 * 2 * 10 = 81; x^2 = (-1 ± √81) / 4 = (-1 ± 9) / 4;

a) x^2 = (-1 - 9) / 4 = - 10/4 = - 5/2 < 0, нет решения;

b) x^2 = (-1 + 9) / 4 = 8/4 = 2;

x = ±2; y = ±2.

2)

{2 (x^2 + y^2) - 1 = 0;

{4 (x^4 + y^4) + 2 (x^2 + y^2) = 40; {2 (x^2 + y^2) = 1;

{4 (x^4 + y^4) + 1 = 40; {2 (x^2 + y^2) = 1;

{4 ((x^2 + y^2) ^2 - 2x^2y^2) = 39; {2 (x^2 + y^2) = 1;

{ (2 (x^2 + y^2)) ^2 - 8x^2y^2 = 39; {2 (x^2 + y^2) = 1;

{1 - 8x^2y^2 = 39; {2 (x^2 + y^2) = 1;

{ (2xy) ^2 = - 19, нет решения.

Ответ: (±2; ±2).