Question

(x-2)^3+(x+2)^3=2(x-3) (x^2+3x+9)

Answer 1

(x - 2)³ + (x + 2)³ = 2(x - 3) (x² + 3x + 9)

х³ - 3х²у + 3ху² - у³

х³ - 6х² + 12х - 8 = 2 ( х³ + 3х² + 9х - 3х² - 9х - 6)

х³ - 6х² + 12х - 8 = 2х³ - 12

х³ - 2х³ - 6х² + 12х = -12 + 8

-х³ - 6х² + 12х = -4

-х(х² + 6х - 12) = -4

х₁ = 4         х² + 6х - 12 = -4

                х² + 6х = -4 + 12

                х² + 6х = 8

                х(х + 6) = 8

                х₂ = 8         х₃ = 2

Answer 2

Let's solve the equation step by step:

Given equation:

$(x - 2)^3 + (x + 2)^3 = 2(x - 3)(x^2 + 3x + 9)$

First, let's expand the cubes and simplify the equation:

$(x^3 - 6x^2 + 12x - 8) + (x^3 + 6x^2 + 12x + 8) = 2(x^3 - 3x^2 + 9x^2 - 27x + 9x + 27)$

Simplify each term on both sides of the equation:

$2x^3 + 24x = 2(6x^2 - 18x + 36)$

Distribute the 2 on the right side:

$2x^3 + 24x = 12x^2 - 36x + 72$

Now, move all terms to one side of the equation:

$2x^3 - 12x^2 + 60x - 72 = 0$

Divide the entire equation by 2 to simplify it:

$x^3 - 6x^2 + 30x - 36 = 0$

Now, let's try to factor this polynomial equation. Since there is no obvious rational root, we'll try to find roots numerically or by using a computer algebra system.

Upon finding the roots, you can verify that they indeed satisfy the original equation. Keep in mind that the solutions might involve complex numbers, as cubic equations can have complex roots. If you're interested in numerical approximations of the roots, you can use numerical methods such as the Newton-Raphson method or utilize computer algebra systems like Mathematica or Wolfram Alpha to find the roots.