Newton's method works like the following. You're given an initial guess for x. Draw the line tangent to the curve at that value of x. Provided the slope of the tangent line isn't zero, the tangent line will intersect the x-axis somewhere. The x-coordinate of the point where the tangent line intersects the x-axis becomes your new guess for x. Draw the line tangent to the curve at the new guess for x, and repeat. The idea is typically the tangent line intersects the x-axis somewhere close to the point where the curve intersects the x-axis.

If the slope of the tangent line at your initial guess is zero (see for example 4(b)), this isn't going to work, and you need to try a different initial guess.

Also, your initial guess for x may lead you in the wrong direction, see for example 4(a).