Find the smallest root of an equation x^3-6x^2+11x-6=0 using ramanujan method 

To find the smallest root of the equation \( x^3 - 6x^2 + 11x - 6 = 0 \) using the Ramanujan method, let's first briefly outline this method.

Ramanujan's approach to finding the roots of a cubic equation involves approximating the smallest real root through successive iterations. The general cubic equation in standard form is:

\[
ax^3 + bx^2 + cx + d = 0
\]

Given that the equation provided is already in this form:

\[
x^3 - 6x^2 + 11x - 6 = 0
\]

We have \( a = 1 \), \( b = -6 \), \( c = 11 \), and \( d = -6 \).

### Step 1: Initial Approximation for Smallest Root

Ramanujan's method suggests that for a cubic equation of this form, an initial approximation for the smallest real root \( x_1 \) is:

\[
x_1 \approx -\left( \frac{d}{a} \right)^{1/3}
\]

Substituting the values of \( d = -6 \) and \( a = 1 \):

\[
x_1 \approx -(-6)^{1/3} = 6^{1/3} \approx 1.817
\]

So, the initial approximation for the smallest root is about \( x_1 \approx 1.817 \).

### Step 2: Refining the Approximation

Using this initial approximation, we can refine the value by solving the equation using numerical methods (like Newton-Raphson method), or we can stop at this approximation if the accuracy is sufficient for practical purposes.

However, the exact root will depend on further iteration or exact methods of solving. For now, the smallest root is approximately \( 1.817 \).

Do you want to see more details or clarifications on this method?

Here are 5 related questions:
1. What are the exact roots of the cubic equation?
2. How does Ramanujan's method differ from other root-finding methods?
3. Can we use the Newton-Raphson method to further refine the root?
4. What are the geometric interpretations of cubic roots?
5. How can we apply Ramanujan’s method to higher-degree polynomials?

**Tip:** Ramanujan's method is particularly useful when an initial estimate for the root is needed quickly without complex calculations.

Find the smallest root of an equation x^3 - 6x^2 + 11x - 6 = 0 using Ramanujan method.

Explore how to find the smallest root of the cubic equation x^3 - 6x^2 + 11x - 6 = 0 using Ramanujan's method. This detailed solution includes an initial approximation formula for root-finding and further insights into refining the root. Suitable for advanced high school or college students.

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