1. **State the problem:** We want to solve the equation $$2x^2 + 3x = 4$$ using an iterative method and find the solution near $$x = -2$$ correct to 4 significant figures. 2. **Rewrite the equation:** Rearrange the equation to isolate $$x$$ in a form suitable for iteration. One way is to write: $$2x^2 + 3x - 4 = 0$$ 3. **Derive an iterative formula:** We can express $$x$$ as: $$x = \frac{4 - 2x^2}{3}$$ This gives the iterative formula: $$x_{n+1} = \frac{4 - 2x_n^2}{3}$$ 4. **Explain iteration:** Starting with an initial guess $$x_0 = -2$$, we compute successive values using the formula until the values converge to 4 significant figures. 5. **Perform iterations:** - $$x_1 = \frac{4 - 2(-2)^2}{3} = \frac{4 - 8}{3} = \frac{-4}{3} = -1.3333$$ - $$x_2 = \frac{4 - 2(-1.3333)^2}{3} = \frac{4 - 2(1.7778)}{3} = \frac{4 - 3.5556}{3} = \frac{0.4444}{3} = 0.1481$$ - $$x_3 = \frac{4 - 2(0.1481)^2}{3} = \frac{4 - 2(0.0219)}{3} = \frac{4 - 0.0438}{3} = \frac{3.9562}{3} = 1.3187$$ - $$x_4 = \frac{4 - 2(1.3187)^2}{3} = \frac{4 - 2(1.739)}{3} = \frac{4 - 3.478}{3} = \frac{0.522}{3} = 0.1740$$ - $$x_5 = \frac{4 - 2(0.1740)^2}{3} = \frac{4 - 2(0.0303)}{3} = \frac{4 - 0.0606}{3} = \frac{3.9394}{3} = 1.3131$$ The values oscillate and do not converge near $$-2$$, so we try another rearrangement. 6. **Alternative iterative formula:** Rearrange as: $$x = \pm \sqrt{\frac{4 - 3x}{2}}$$ Since we want a solution near $$-2$$, choose the negative root: $$x_{n+1} = -\sqrt{\frac{4 - 3x_n}{2}}$$ 7. **Perform iterations with new formula:** Starting with $$x_0 = -2$$: - $$x_1 = -\sqrt{\frac{4 - 3(-2)}{2}} = -\sqrt{\frac{4 + 6}{2}} = -\sqrt{5} = -2.2361$$ - $$x_2 = -\sqrt{\frac{4 - 3(-2.2361)}{2}} = -\sqrt{\frac{4 + 6.7083}{2}} = -\sqrt{5.3542} = -2.3135$$ - $$x_3 = -\sqrt{\frac{4 - 3(-2.3135)}{2}} = -\sqrt{\frac{4 + 6.9405}{2}} = -\sqrt{5.4703} = -2.3390$$ - $$x_4 = -\sqrt{\frac{4 - 3(-2.3390)}{2}} = -\sqrt{\frac{4 + 7.0170}{2}} = -\sqrt{5.5085} = -2.3466$$ - $$x_5 = -\sqrt{\frac{4 - 3(-2.3466)}{2}} = -\sqrt{\frac{4 + 7.0398}{2}} = -\sqrt{5.5199} = -2.3490$$ - $$x_6 = -\sqrt{\frac{4 - 3(-2.3490)}{2}} = -\sqrt{\frac{4 + 7.0470}{2}} = -\sqrt{5.5235} = -2.3497$$ 8. **Check convergence:** The values are converging near $$-2.3497$$. 9. **Final answer:** The solution near $$x = -2$$ correct to 4 significant figures is: $$\boxed{-2.350}$$