# Mathematical equations can be solved in many ways; however some equations cannot be solved algebraically. I am going to show the three methods of solving these types of equations numerically

| Page                Milan Vashi

Maths Core 3: Coursework

## Introduction

Mathematical equations can be solved in many ways; however some equations cannot be solved algebraically. I am going to show the three methods of solving these types of equations numerically, also I will show each method working, and each method failing.

The three methods are:

• Change Of Sign Method
• Rearranging  f(x)= 0 Into The Form x=g(x)
• Newton-Raphson Method

## Change of Sign Method

### Change of Sign Method Working

I am going to solve the equation f(x) = 0, where f(x) =. The graph of y=f(x) is shown here.

In order to calculate the value of the root in [0.3, 0.4] to 3 decimal places I must check whether the value is closer to 0.309 than 0.310 meaning that the value 0.3095 needs to be calculated. If this value is negative then the root is 0.309 to 3 decimal places however if the value is positive then the root would have to be given as 0.310.

The value of f(x) when x=0.3095 is 0.00614708 meaning that I can deduce that the value of the root in [0, 1] to 3 decimal places is 0.310.

The error bounds are 0.3095 and 0.310.

## Change of Sign Method Failing

Consider the equation f(x) =0, where f(x) = I shall try to locate the roots by making a table of values that correspond to this function of x.

## Rearranging f(x) =0 into x=g(x)

### Rearranging f(x) =0 into x=g(x) failing

I am going to solve the equation f(x) =0 where f(x) =. The graph of y=f(x) is shown below.

•

The sequence formed is converging; however it converges to the incorrect root meaning that it is considered as a failure. , therefore . As we know that the root <-3, g’(x)>1 meaning that it fails as the range which enables the method to succeed -1<g’(x) <1 is not fulfilled by this equation.

### Rearranging f(x) =0 into x=g(x) working

I am going to solve the equation f(x) =0 where f(x) =. The graph of y=f(x) is shown below.

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