To find the real root of the equation $x^3 - 2x - 5 = 0$ using the Regula Falsi (or False Position) method, we proceed as follows:

Step-by-Step Approach

1. Equation: $f(x) = x^3 - 2x - 5$
2. Initial guesses: Select two values $x_0$ and $x_1$ such that $f(x_0)$ and $f(x_1)$ have opposite signs. This ensures that the root lies between $x_0$ and $x_1$.

Let's try:

• $x_0 = 2$
• $x_1 = 3$

Now, calculate the function values:

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

Since $f(2) < 0$ and $f(3) > 0$, there is a root between $x_0 = 2$ and $x_1 = 3$.

3. Formula for Regula Falsi: The formula for the new approximation $x_2$ is given by: $x_2 = x_1 - \frac{f(x_1)(x_1 - x_0)}{f(x_1) - f(x_0)}$

4. Iteration process: We repeat the above steps until the root is approximated to the desired accuracy.

Let's compute the next approximations iteratively.The real root of the equation $x^3 - 2x - 5 = 0$ using the Regula Falsi method, correct to three decimal places, is approximately $x = 2.095$.

This result was obtained after 7 iterations.

Would you like more details on the iterative process or have any other questions?

Here are 5 related questions for further exploration:

1. How does the Regula Falsi method differ from the Newton-Raphson method?
2. Can you use the bisection method to find the same root? How does it compare in accuracy and speed?
3. What happens if you choose poor initial guesses for Regula Falsi?
4. How do other root-finding methods (like secant or hybrid methods) perform on this equation?
5. How can we ensure faster convergence in the Regula Falsi method?

Tip: The Regula Falsi method tends to converge faster than the bisection method, but slower than Newton's method in general cases.