1. **State the problem:** Solve the equation $1.1 \times 10^{-12} = 2x^2(x + 0.1)$ for $x$. 2. **Rewrite the equation:** $$1.1 \times 10^{-12} = 2x^2(x + 0.1)$$ 3. **Expand the right side:** $$2x^2(x + 0.1) = 2x^3 + 0.2x^2$$ 4. **Rewrite the equation as:** $$2x^3 + 0.2x^2 = 1.1 \times 10^{-12}$$ 5. **Bring all terms to one side:** $$2x^3 + 0.2x^2 - 1.1 \times 10^{-12} = 0$$ 6. **Divide the entire equation by 2 to simplify:** $$\frac{2x^3}{2} + \frac{0.2x^2}{2} - \frac{1.1 \times 10^{-12}}{2} = 0$$ $$x^3 + 0.1x^2 - 5.5 \times 10^{-13} = 0$$ 7. **Since the constant term is very small, try to find approximate roots by inspection or numerical methods.** 8. **Try $x \approx 10^{-4}$:** $$x^3 = (10^{-4})^3 = 10^{-12}$$ $$0.1x^2 = 0.1 \times (10^{-4})^2 = 0.1 \times 10^{-8} = 10^{-9}$$ Sum: $10^{-12} + 10^{-9} = 1.001 \times 10^{-9}$ which is much larger than $5.5 \times 10^{-13}$, so try smaller $x$. 9. **Try $x \approx 10^{-5}$:** $$x^3 = (10^{-5})^3 = 10^{-15}$$ $$0.1x^2 = 0.1 \times (10^{-5})^2 = 0.1 \times 10^{-10} = 10^{-11}$$ Sum: $10^{-15} + 10^{-11} = 1.0001 \times 10^{-11}$ still larger than $5.5 \times 10^{-13}$. 10. **Try $x \approx 10^{-6}$:** $$x^3 = (10^{-6})^3 = 10^{-18}$$ $$0.1x^2 = 0.1 \times (10^{-6})^2 = 0.1 \times 10^{-12} = 10^{-13}$$ Sum: $10^{-18} + 10^{-13} = 1.00001 \times 10^{-13}$ which is less than $5.5 \times 10^{-13}$. 11. **By interpolation, the root is approximately between $10^{-6}$ and $10^{-5}$.** 12. **Use numerical methods (e.g., Newton-Raphson) for precise root, but approximate solution is:** $$x \approx 7 \times 10^{-6}$$ **Final answer:** $$x \approx 7 \times 10^{-6}$$