1. **State the problem:** We want to find an approximate solution to the equation $$x^3 - 5x^2 - 12 = 0$$ using the iterative formula $$x_{n+1} = 5 + \frac{12}{x_n^2}$$ starting with $$x_1 = 5$$. 2. **Explain the iterative formula:** This formula is derived by rearranging the original equation to isolate $$x$$ on one side. We use the previous value $$x_n$$ to calculate the next value $$x_{n+1}$$. 3. **Calculate successive iterations:** - Start with $$x_1 = 5$$. - Calculate $$x_2$$: $$x_2 = 5 + \frac{12}{5^2} = 5 + \frac{12}{25} = 5 + 0.48 = 5.48$$ - Calculate $$x_3$$: $$x_3 = 5 + \frac{12}{(5.48)^2} = 5 + \frac{12}{30.0304} \approx 5 + 0.3996 = 5.3996$$ - Calculate $$x_4$$: $$x_4 = 5 + \frac{12}{(5.3996)^2} = 5 + \frac{12}{29.157} \approx 5 + 0.4117 = 5.4117$$ - Calculate $$x_5$$: $$x_5 = 5 + \frac{12}{(5.4117)^2} = 5 + \frac{12}{29.292} \approx 5 + 0.4097 = 5.4097$$ 4. **Check for convergence:** The values are stabilizing around $$5.41$$. 5. **Final answer:** To 2 decimal places, the approximate solution is $$\boxed{5.41}$$.