x³ - 2x - 18 = 0 → let: x = u + v

(u + v)³ - 2.(u + v) - 18 = 0

[(u + v)².(u + v)] - 2.(u + v) - 18 = 0

[(u² + 2uv + v²).(u + v)] - 2.(u + v) - 18 = 0

[u³ + u²v + 2u²v + 2uv² + uv² + v³] - 2.(u + v) - 18 = 0

[u³ + v³ + 3u²v + 3uv²] - 2.(u + v) - 18 = 0

[(u³ + v³) + (3u²v + 3uv²)] - 2.(u + v) - 18 = 0

[(u³ + v³) + 3uv.(u + v)] - 2.(u + v) - 18 = 0

(u³ + v³) + 3uv.(u + v) - 2.(u + v) - 18 = 0 → you can factorize: (u + v)

(u³ + v³) + (u + v).(3uv - 2) - 18 = 0 → suppose that: (3uv - 2) = 0 ← equation (1)

(u³ + v³) + (u + v).(0) - 18 = 0

(u³ + v³) - 18 = 0 ← equation (2)

You can get a system of 2 equations:

(1) : (3uv - 2) = 0

(1) : 3uv = 2

(1) : uv = 2/3

(1) : u³v³ = 8/27

(2) : (u³ + v³) - 18 = 0

(2) : u³ + v³ = 18

Let: U = u³

Let: V = v³

You can get a new system of 2 equations:

(1) : UV = 8/27 ← this is the product P

(2) : U + V = 18 ← this is the sum S

You know that the values U & V are the solutions of the following equation:

x² - Sx + P = 0

x² - 18x + (8/27) = 0

Δ = (- 18)² - [4 * (8/27)]

Δ = 8716/27

Δ = 4 * (2179/27)

x₁ = [18 + 2√(2179/27)]/2 = 9 + √(2179/27) ← this is U

x₂ = [18 - 2√(2179/27)]/2 = 9 - √(2179/27) ← this is V

Recall: u³ = U → u = U^(1/3) = [9 + √(2179/27)]^(1/3)

Recall: v³ = V → v = V^(1/3) = [9 - √(2179/27)]^(1/3)

Recall: x = u + v

x = (u + v)

x = [9 + √(2179/27)]^(1/3) + [9 - √(2179/27)]^(1/3)

x ≈ 2.87440013802963

f(x)=x^3-2x-18=>f '(x)=3x^2-2=>x(n)=x(n-1)-f(x(n-1))/f '(x(n-1))=>x1=x(0)-f(x0)/f '(x0), where x0=3=>x(1)=3-3/25=2.88=>

x2=2.88-0.1278725/22.8832=2.874412=>x3=2.874412-.0002708435/22.78673=2.8744.

Ans. x3=2.8744.