The problem involves solving the equation $2x^3 - 8x^2 + 10x - 4 = 0$ using an iterative formula. Let's proceed step by step:

Iteration Formula:

The given iteration formula is: $x_{n+1} = \sqrt[3]{\frac{8(x_n)^2 - 10x_n + 4}{2}}$

Step-by-Step Solution:

1. Start with $x_0 = 0$ (initial value).

2. Calculate $x_1$: $x_1 = \sqrt[3]{\frac{8(0)^2 - 10(0) + 4}{2}}$ Simplify: [ x_1 = \sqrt[3]{\frac{4}{2}} = \sqrt[3]{2} ] Using a calculator, $\sqrt[3]{2} \approx 1.260$.

3. Calculate $x_2$: Substitute $x_1 = 1.260$ into the formula: $x_2 = \sqrt[3]{\frac{8(1.260)^2 - 10(1.260) + 4}{2}}$ First, compute the numerator: $8(1.260)^2 = 8(1.5876) = 12.7008, \quad -10(1.260) = -12.6, \quad +4.$ So, the numerator is: $12.7008 - 12.6 + 4 = 4.1008.$ Divide by 2: $\frac{4.1008}{2} = 2.0504.$ Take the cube root: $x_2 = \sqrt[3]{2.0504} \approx 1.276 \, (\text{to 3 decimal places}).$

Final Answer:

$x_2 = 1.276$

Would you like me to break down any part of this solution further? 😊

Related Questions:

1. What is the purpose of iteration in numerical methods?
2. Can this method always guarantee convergence to a root?
3. What is the significance of rearranging the equation for iteration?
4. How does the choice of $x_0$ affect the result?
5. Could this equation have more than one real root?

Tip:

When solving iteratively, always round intermediate steps only after the final result to ensure accuracy.