Решите уравнение 1) (x+1)^3-(x-1)^3=x(6x+2) 2)(x+2)^3-(x-1)^3=9x^2+36 3)(x-2)^3-3x^2-4=(x-3)^3 4)(x+3)^3=x^2(x+9)

1) (х + 1)³ – (х – 1)³ = х(6х + 2);

(х³ +3х² * 1 + 3х * 1² + 1³) – (х³ – 3х² * 1 + 3х * 1² – 1³) = 6х² + 2;

х³ +3х² + 3х + 1 – х³ + 3х² - 3х + 1 = 6х² + 2;

х³ - х³ + 3х² + 3х² – 6х² + 3х - 3х – 2х + 1 + 1 = 0;

-2х + 2 = 0;

-2х = -2;

2х = 2;

х = 2 : 2;

х = 1.

Проверка:

(1 + 1)³ – (1 – 1)³ = 1 * (6 * 1 + 2);

2³ – 0³ = 6 + 2;

8 = 8.

2) (x + 2)³ - (x - 1)³ = 9x² + 36;

(x³ + 3x² * 2 + 3x * 2² + 2³) – (x³ – 3x² * 1 + 3x * 1² – 1³) = 9x² + 36;

(x³ + 6x² + 3x * 4 + 8) – (x³ – 3x² + 3x – 1) = 9x² + 36;

x³ + 6x² + 12x + 8 – x³ + 3x² - 3x + 1 - 9x² – 36 = 0;

x³ – x³ + 6x² + 3x² - 9x² + 12x - 3x + 8 + 1  – 36 = 0;

9х – 27 = 0;

9х = 27;

х = 27 : 9;

х = 3.

Проверка:

(3 + 2)³ - (3 - 1)³ = 9 * 3² + 36;

5³ – 2³ = 81 + 36;

125 – 8 = 117;

117 = 117.

3) (x - 2)³ - 3x² – 4 = (x - 3)³;

x³ – 3x² * 2 + 3х * 2² – 2³ – 3х² – 4 = x³ – 3х² * 3 + 3х * 3² – 3³;

x³ – 6x² + 3х * 4 – 8 – 3х² – 4 = x³ – 9х² + 3х * 9 – 27;

x³ – 6x² + 12х – 8 – 3х² – 4 = x³ – 9х² + 27х – 27;

x³ – x³ - 6x² – 3х² + 9х² + 12х - 27х – 8  – 4 + 27 = 0;

-15х + 15 = 0;

-15х = -15;

15х = 15;

х = 15 : 15;

х = 1.

Проверка:

(1 - 2)³ – 3 * 1² – 4 = (1 - 3)³;

-1³ – 3 – 4 = -2³;

-1 – 3 – 4 = -8;

-8 = -8.

4) (x + 3)³ = x²(x + 9);

х³ + 3 * х² * 3 + 3x * 3² + 3³ = x³ + 9x²;

х³ + 9х² + 3x * 9 + 27 = x³ + 9x²;

х³ - x³ + 9х² - 9x² + 27x + 27 = 0;

27х + 27 = 0;

27х = -27;

х = -27 : 27;

х = -1.

Проверка:

(-1 + 3)³ = (-1)² * (-1 + 9);

2³ = 1 * 8;

8 = 8.