 # Introduction

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## Introduction

#### 'Normal' factorising techniques

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

For example:

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

Let's look at an example to show this:

Solve: x3 = x2 + 7x + 20

Firstly rewrite the equation as f(x) = 0

Therefore: x3 - x2 - 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 x3 - x2 - 7x - 20 can now be written as...

x3 - x2 - 7x - 20

= (x − 4)(x2 + bx + c)

= x3 + (b − 4)x2 + (c − 4b)x − 4c

Matching the expressions gives us:

b − 4 = -1, so b = 3, and

c − 4b = -7, so c = 5.

Therefore:

x3 - x2 - 7x - 20 = (x − 4)(x2 + 3x + 5)

The quadratic in this case does not factorise (b2 - 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

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

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