(х+1)(х+2)(х+3)(х+4)=1 РЕШИТЬ УРАВНЕНИЕ

(х + 1)(х + 2)(х + 3)(х + 4) = 1;

(х + 1)(х + 4) * (х + 2)(х + 3) = 1;

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

(x^2 + 5x + 4)(x^2 + 5x + 6) = 1;

Пусть x^2 + 5x + 4 = a, тогда x^2 + 5x + 6 = a + 2;

a(a + 2) = 1;

a^2 + 2a - 1 = 0;

D = 4 + 4 = 8;

a1,2 = (-2 +- sqrt8)/2 = (-2 +- 2sqrt2)/2 = -1 +- sqrt2;

a1 = -1 - sqrt2, a2 = -1 + sqrt2;

x^2 + 5x + 4 = -1 - sqrt2;

x^2 + 5x + 4 + 1 + sqrt2 = 0;

x^2 + 5x + (5 + sqrt2) = 0;

D = 25 - 4 * (5 + sqrt2) = 25 - 20 - 4sqrt2 = 5 - 4sqrt2 < 0 - нет решений;

x^2 + 5x + 4 = -1 + sqrt2;

x^2 + 5x + (5 - sqrt2) = 0;

D = 25 - 4 * (5 - sqrt2) = 25 - 20 + 4sqrt2 = 5 + 4sqrt2;

x = (-5 +- sqrt(5 + 4sqrt2))/2;

x1 = (-5 + sqrt(5 + 4sqrt2))/2;

x2 = (-5 - sqrt(5 + 4sqrt2))/2.

Ответ: (-5 + sqrt(5 + 4sqrt2))/2; (-5 - sqrt(5 + 4sqrt2))/2.