1. **State the problem:** We want to estimate the solution to the equation $$x^3 + 7x - 5 = 0$$ between $$x=0$$ and $$x=1$$ using the iteration formula $$x_{n+1} = \frac{5}{x_n^2 + 7}$$ starting with $$x_0 = 1$$. 2. **Explain the iteration formula:** This formula is derived by rearranging the original equation to isolate $$x$$ on one side. We use it to generate successive approximations $$x_1, x_2, x_3$$ that get closer to the actual root. 3. **Perform the iterations:** - Calculate $$x_1 = \frac{5}{1^2 + 7} = \frac{5}{8} = 0.6250$$ - Calculate $$x_2 = \frac{5}{(0.6250)^2 + 7} = \frac{5}{0.390625 + 7} = \frac{5}{7.390625} \approx 0.6767$$ - Calculate $$x_3 = \frac{5}{(0.6767)^2 + 7} = \frac{5}{0.4579 + 7} = \frac{5}{7.4579} \approx 0.6704$$ 4. **Final estimate:** After three iterations, the estimate for the solution is $$\boxed{0.6704}$$ correct to 4 decimal places. This iterative method converges to the root between 0 and 1 of the given cubic equation.