Find the root for 2x³-6x²+11x-7 using ramanujan method

To find the root of the cubic equation \( 2x^3 - 6x^2 + 11x - 7 = 0 \) using Ramanujan's method, we follow these steps:

### Step-by-Step Process

1. **Write the cubic equation:**
   \[
   2x^3 - 6x^2 + 11x - 7 = 0
   \]
   Divide the equation by the leading coefficient (2):
   \[
   x^3 - 3x^2 + \frac{11}{2}x - \frac{7}{2} = 0
   \]

2. **Initial approximation of the root**: Ramanujan's method starts with an approximation. Let's assume \( x = 2 \) as a trial root based on visual inspection or rough estimates.

3. **Check \( f(x) \) at \( x = 2 \):**
   \[
   f(2) = 2(2)^3 - 6(2)^2 + 11(2) - 7 = 16 - 24 + 22 - 7 = 7
   \]
   So, \( x = 2 \) is not a root, but it can serve as a good starting point.

4. **Refine the approximation using Ramanujan's method:**
   The method uses the formula:
   \[
   x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}
   \]
   We first need to compute the derivative of the function:
   \[
   f'(x) = 6x^2 - 12x + 11
   \]

5. **Calculate \( f'(2) \):**
   \[
   f'(2) = 6(2)^2 - 12(2) + 11 = 24 - 24 + 11 = 11
   \]

6. **Apply the approximation formula:**
   \[
   x_1 = 2 - \frac{7}{11} = 2 - 0.6364 \approx 1.3636
   \]

7. **Check \( f(x_1) \) at \( x_1 \approx 1.3636 \):**
   \[
   f(1.3636) = 2(1.3636)^3 - 6(1.3636)^2 + 11(1.3636) - 7
   \]
   Performing this calculation will provide a more refined result, but the approximation already improves with \( x_1 \approx 1.3636 \).

Thus, using Ramanujan's iterative method, the first approximation of the root is \( x \approx 1.3636 \).

Would you like a detailed breakdown of the remaining steps, or any further clarifications?

---

### Related Questions:
1. What is Ramanujan's iterative method and how does it differ from Newton's method?
2. How do you determine a good initial guess for cubic equations?
3. Can Ramanujan's method always find all roots for a cubic equation?
4. How can we calculate the remaining roots for \( 2x^3 - 6x^2 + 11x - 7 = 0 \)?
5. What are the limitations of using iterative methods for root finding?

### Tip:
Always ensure that the initial guess in iterative methods is close to the actual root for faster convergence!

Find the root for 2x³-6x²+11x-7 using Ramanujan method

Learn how to find the root of the cubic equation 2x³-6x²+11x-7 using Ramanujan's iterative method. This detailed guide covers step-by-step solutions, derivative calculations, and provides insights into advanced algebraic techniques. Suitable for advanced high school or undergraduate students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation