Решите уравнение (3 х^2-x-2) ^2 + (6x^2+x-1) ^4=0

(3 * х^2 - x - 2) ^2 + (6 * x^2 + x - 1) ^4 = 0;

1) 3 * x^2 - x - 2 = 0;

D = b^2 - 4 * a * c = (-1) ^2 - 4 * 3 * (-2) = 1 + 12 * 2 = 1 + 24 = 25 = 5^2;

x1 = ( - (-1) + 5) / (2 * 3) = (1 + 5) / 6 = 6/6 = 1;

x2 = ( - (-1) - 5) / (2 * 3) = (1 - 5) / 6 = - 4/6 = - 2/3;

2) 6 * x^2 + x - 1 = 0;

D = 1^2 - 4 * 6 * (-1) = 1 + 24 * 1 = 1 + 24 = 25 = 5^2;

x1 = (-1 + 5) / (2 * 6) = 4/12 = 1/3;

x2 = (-1 - 5) / (2 * 6) = - 6/12 = - 1/2;

Отсюда получаем:

(3 * (x - 1/3) * (x + ½)) ^2 + (6 * (x - 1/3) * (x + ½)) ^4 = 0;

3 * 3 * (x - 1/3) ^2 * (x + ½) ^2 + 6 * 6 * (x - 1/3) ^4 * (x + ½) ^4 = 0;

(x - 1/3) ^2 * (x + ½) ^2 * (9 + 36 * (x - 1/3) ^2 * (x + ½) ^2) = 0;

(x - 1/3) ^2 * (x + ½) ^2 * (9 + 3 * 3 * 2 * 2 * (x - 1/3) * (x - 1/3) * (x + ½) * (x + ½)) = 0;

(x - 1/3) ^2 * (x + ½) ^2 * (9 + (3 * x - 1) ^2 * (2 * x + 1) ^2) = 0;

1) 9 + (3 * x - 1) ^2 * (2 * x + 1) ^2 = 0;

Уравнение не имеет корней.

2) (x - 1/3) ^2 = 0;

x - 1/3 = 0;

x = 1/3;

3) (x + ½) ^2 = 0;

x + ½ = 0;

x = - 1/2;

Ответ: х = 1/3 и х = - 1/2.