The secant method is an iterative numerical technique used to find roots of a nonlinear equation. It approximates the root of a function using a sequence of secant lines.

Given Equation:

$f(x) = x^3 - 2x - 5$ We need to find the root of this equation (i.e., find $x$ such that $f(x) = 0$).

Secant Method Formula:

The iteration formula for the secant method is: $x_{n+1} = x_n - \frac{f(x_n)(x_n - x_{n-1})}{f(x_n) - f(x_{n-1})}$ Where:

• $x_{n+1}$ is the next approximation,
• $x_n$ and $x_{n-1}$ are two previous approximations.

Steps to Solve:

1. Choose two initial guesses: $x_0$ and $x_1$.
2. Iterate using the secant formula until the root is found to the desired accuracy.

Let's go through the steps for this specific equation $f(x) = x^3 - 2x - 5$.

Step-by-Step Process:

1. Choose Initial Guesses: Let’s choose $x_0 = 2$ and $x_1 = 3$.

2. Evaluate $f(x)$:

   • $f(2) = 2^3 - 2(2) - 5 = 8 - 4 - 5 = -1$,
   • $f(3) = 3^3 - 2(3) - 5 = 27 - 6 - 5 = 16$.

3. Apply Secant Formula: $x_2 = 3 - \frac{16(3 - 2)}{16 - (-1)} = 3 - \frac{16}{17} \approx 2.0588$

4. Next Iteration: Now take $x_1 = 2.0588$ and $x_0 = 3$, and repeat the process to find the next approximation.

Would you like me to carry out more iterations, or explain the steps further?

Here are some related questions:

1. What are other methods to solve nonlinear equations?
2. How does the secant method compare to Newton’s method?
3. Can the secant method fail, and under what conditions?
4. How do convergence properties affect numerical methods like the secant method?
5. How do initial guesses influence the secant method?

Tip: The secant method doesn't require the derivative of the function, unlike Newton's method, making it useful when derivatives are difficult to compute.