1. The problem asks when Newton's Method fails to find an approximate solution of an equation. 2. Newton's Method uses the formula: $$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$$ where $f'(x_n)$ is the slope of the tangent line at $x_n$. 3. For Newton's Method to work, the slope $f'(x_n)$ must not be zero because division by zero is undefined. 4. If the tangent line at any approximated point is parallel to the x-axis, then $f'(x_n) = 0$. 5. Therefore, Newton's Method fails if the tangent line at any approximated point is parallel to the x-axis. Final answer: Newton's Method fails if the tangent line (at any approximated point) is parallel to the x-axis.