# 2. Newton's Method for Solving Equations

by M. Bourne

Computers use **iterative methods** to solve
equations. The process involves making a guess at the true
solution and then applying a formula to get a better guess and so
on until we arrive at an acceptable approximation for the
solution.

If we wish to find *x* so that `f(x)
= 0` (a common type of problem), then we guess some initial value *x*_{0} which is close to the desired solution and then we get a better approximation using **Newton's Method**:

`x_1=x_0-(f(x_0))/(f'(x_0)`

[This is just based on the point-slope form of a straight line].

### Example 1

### Need Graph Paper?

Find the root of

2

x^{2}−x− 2 = 0

between `1` and `2`.

This example has another root, which is negative.

You can explore this example further in Newton's Method Interactive Graph.

## Non-polynomial Functions with Multiple Roots

When using a computer to find roots of more complicated functions it's best to **understand** what is going on and give the computer a guess close to your desired answer.

### Example 2

Solve 1− *t*^{2} + 2^{t} = 0

[Certain math software is not able to find the solution directly for us. We need to know how to properly use the tool to get the solution, either with graphs or setting up Newton's Method. This could involve giving an initial estimate for the root.]

## Further Examples

### Example 3

Solve 3*x*^{3} − 9*x*^{2} + 5*x* + 2 = 0 using Newton's Method.

### Example 4

Solve *x*^{2} = 0 using Newton's Method.

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