### Introduction

In this Section we briefly discuss nonlinear equations (what they are and what their solutions might be) before noting that many such equations which crop up in applications cannot be solved exactly.

The remainder (and majority) of the Section then goes on to discuss methods for approximating solutions of nonlinear equations.

#### Prerequisites

- understand derivatives of simple functions
- understand the quadratic formula
- understand exponentials and logarithms

#### Learning Outcomes

- approximate roots of equations by the bisection method and by the Newton-Raphson method
- implement an approximate Newton-Raphson method

#### Contents

1 Nonlinear Equations2 The bisection method

2.1 Pros and cons of the bisection method

3 The Newton-Raphson method

4 Engineering Example 2

4.1 Pressure in an ideal multi-component mixture

4.2 An approximate Newton-Raphson method