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# Introduction

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## Introduction

Many types of equations can be solved using 'normal' techniques.

For example:

can be solved by**Linear equations****rearrangement**can be solved using the**Quadratic equations****quadratic formula**can be solved by**More complex equations****factorising by inspection**, or by using the**Factor Theorem**.

Let's look at an example to show this:

Solve: x^{3} = x^{2} + 7x + 20

Firstly rewrite the equation as f(x) = 0

Therefore: x^{3} - x^{2} - 7x - 20 = 0

This has no obvious factors, so use the factor theorem. If f(a) = 0 then (x − a) is a factor. We then try values for a.

f(4) = 64 - 16 - 28 - 20 = 0. Therefore (x - 4) is a factor.

To find the other solutions we know that x^{3} - x^{2} - 7x - 20 can now be written as...

x^{3} - x^{2} - 7x - 20

= (x − 4)(x^{2} + bx + c)

= x^{3} + (b − 4)x^{2} + (c − 4b)x − 4c

**Matching the expressions gives us:**

b − 4 = -1, so b = 3, and

c − 4b = -7, so c = 5.

Therefore:

x^{3} - x^{2} - 7x - 20 = (x − 4)(x^{2} + 3x + 5)

The quadratic in this case does not factorise (b^{2} - 4ac < 0). So there is only one solution to the equation: x = 4.

When this technique does not work we need a new method. **Numerical Methods** are methods that can be used in these cases.

**Numerical Methods** are systems, or algorithms, for solving equations that cannot be solved using normal techniques. There are a number of different types of numerical methods available. The ones you need for your exams are listed below and are shown in more detail in the next Learn-It:

- Change of the Sign Methods
- Newton Raphson
- Rearrangement