# Rearrangement

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## Rearrangement

This is an alternative method that rearranges the original question into two equations, a straight line and a curve, and then finds where these meet. **It works as follows:**

- Rearrange the equation to be solved into the form x = g(x). [g(x) is a different function].
- The solution to this equation is to find where y = x meets y = g(x). This is where the coordinates on g(x) are the same.
- In this situation, we can guess a value x
_{0}and hope that g(x_{0}) will be a better guess. This can be iterated repeatedly.

Below is a diagram which shows that by starting at a value of x_{0} and repeating the iterations (i.e. finding x_{1}, x_{2},...), you get closer and closer to the point where the two lines cross.

**Performing this iteration produces two possible results:**

- It diverges (i.e. it gets further and further away from the start). This means the rearrangement has not worked.
- It converges (i.e. it homes in) to the root, and solves the equation.

**Example:**

Let's take the example used in the Newton Raphson Learn-It.

We want to find the roots to the following equation:

f(x) = e^{x} − 3x,

This can be rearranged to x = e^{x} ⁄ 3

Let's estimate that the root to this equation is 0.6.

Then taking the **iteration equation** above [x_{n+1} = g(x_{n})]
and substituting our equation gives:

**Performing this iteration a number of times gives:**

X_{1} | X_{2} | X_{3} | X_{4} | X_{5} | X_{6} | X_{7} | X_{8} | X_{9} | X_{10} |

0.607 | 0.612 | 0.615 | 0.616 | 0.617 | 0.618 | 0.618 | 0.619 | 0.619 | 0.619 |

As the solution gives consistent values of 0.619, this suggests that a root of the equation f(x) = e^{x} − 3x is 0.619 (to 3 decimal places).