# Algebra resources

### Facts & Formulae Leaflets (2)

Differentiation for Economics and Business Studies Functions of Multi-Variable Functions

Overview of the rules of partial differentiation and methods of optimization of functions in Economics and Business Studies. This leaflet has been contributed to the mathcentre Community Project by Morgiane Richard (University of Aberdeen) and reviewed by Anthony Cronin (University College Dublin).

Exponential and Logarithm for Economics and Business Studies

Overview of the properties of the functions e and ln and their applications in Economics. This leaflet has been contributed to the mathcentre Community Project by Morgiane Richard (University of Aberdeen) and reviewed by Shazia Ahmed (University of Glasgow) and Anthony Cronin (University College Dublin).

### Practice & Revision (1)

Algebra Refresher - Interactive version

An interactive version of the refresher booklet on Algebra including links to other resources for further explanation. It includes revision, exercises and solutions on fractions, indices, removing brackets, factorisation, algebraic frations, surds, transpostion of formulae, solving quadratic equations and some polynomial equations, and partial fractions. An interactive version and a welsh language version are available.

### Quick Reference (24)

Factorising quadratics

In this leaflet, we explain the procedure for factorising quadratic expressions. Be aware, not all quadratic expressions can be factorised.

Factorising simple expressions

Factorising can be thought of as a reversal of the process of removing brackets. When we factorize an expression, it is written as a product of two or more terms, and these will normally involve brackets.

Indicial equations

An indicial equation is one in which the power is unknown. Such equations often occur in the calculation of compound interest.

Linear inequalities

The ability to study regions defined by linear inequalities is helpful when studying linear programming. This leaflet reminds you how to sketch these regions.

Linear relationships

In many business applications, two quantities are related linearly. This means a graph of their relationship forms a straight line. This leaflet discusses one form of the mathematical equation which describes linear relationships.

Logarithms

We use logarithms to write expressions involving powers in a different form. If you can work confidently with powers, you should have no problems handling logarithms

Manipulating Inequalities

This leaflet explains the meaning of the inequality symbols < and >, and shows how expressions involving them are manipulated.

Mathematical Symbols and Abbreviations

This leaflet provides information on symbols and notation commonly used in mathematics. It shows the meaning of a symbol and, where necessary, an example and an indication of how the symbol would be said. For further information from mathcentre resources, a search phrase is given. This Quick Reference leaflet is contributed to the mathcentre Community Project by Janette Matthews and reviewed by Tony Croft, University of Loughborough.

Negative and fractional powers

Sometimes it is useful to use negative and fractional powers. These are explained in this leaflet.

Quadratic equations 2

This leaflet will explain how quadratic equations can be solved using a formula.

Rearranging Formulas 1

The ability to rearrange formulas, or rewrite them in different ways, is an important skills. This leaflet will explain how to rearrange some simple formulas.

Rearranging Formulas 2

The ability to rearrange formulas, or rewrite them in different ways, is an important skills. This leaflet will explain how to rearrange some complex formulas.

Removing brackets 1

In order to simplify mathematical expressions, it is frequently necessary to 'remove brackets'. This leaflet explains the rules for replacing bracketed terms, with non-bracketed equivalents.

Removing brackets 2

In this leaflet, we show you the correct procedure for writing expressions of the form (a+b)(c+d) in an alternative form, without brackets.

Sigma notation

Sigma notation provides a concise and convenient way of writing long sums. This leaflet explains how.

Simplifying fractions

Fractions involving symbols occur frequently. It is necessary to be able to simplify these and rewrite them in different, but equivalent forms. In this leaflet, we revise how these processes are carried out.

Simultaneous equations

On occasions you will come across two or more unknown quantities, and two or more equations relating to them. These are called simultaneous equations and when asked to solve them yo umust find values of the unknowns which satisfy all the given equations at the same time. On this leaflet we will illustrate one way in which this can be done.

Solving linear equations

Equations always involve one or more unknown quantities which we try to find when we solve the equation. The simplest equations to deal with are linear equations. On this leaflet we describe how these are solved.

Substitution

This leaflet revised the way in which symbols in formulas are replaced by actual numerical values - a process known as substitution. You will need a calculator to check these examples.

Symbols

Mathematics provides a very rich language for the communication of concepts and ideas, and a set of powerful tools for the solution of problems. In order to use this language, it is essential to appreciate how symbols are used to represent quantities, and to understand the conventions which have been developed to manipulate them.

The exponential constant e

The letter e is used in many mathematical calculations to stand for a particular number known as the exponential constant. This leaflet provides information about this important constant, and the related exponential function.

The laws of indices

When a number is to be multiplied by itself, a power or index can be used to write this compactly. In this leaflet, we remind you of how this is done, and state a number of rules, or laws, which can be used to simplify expressions involving indices.

The laws of logarithms

There are a number of rules known as the laws of logarithms. These allow expressions involving logarithms to be rewritten in a variety of different ways. The laws apply to logarithms of any base, but the same base must be used throughout a calculation.

### Teach Yourself (16)

Completing the square

It is often useful to be able write a quadratic expression in an alternative form - that is as a complete square plus or minus a number. The process for doing this is called completing the square. This booklet explains how this process is carried out.

Completing the square - maxima and minima

This is a workbook which describes how to complete the square for a quadratic expression. It goes on to show how the technique can be used
to find maximum or minimum values of a quadratic expression.

Cubic equations

This booklet explains what is meant by a cubic equation and discusses the nature of the roots of cubic equations.
It explains a process called synthetic division which can be used to locate further roots when one root is known.
The graphical solution of cubic equations is also described.

Expanding, or removing brackets

This is a complete workbook covering the removal of brackets
from expressions. It contains lots of examples and exercises.
It can be used as a free-standing resource, or can be read in conjunction with mathtutor - the companion on-disk resource.

Factorising quadratics

The ability to factorise a quadratic expression is an essential skill.
This booklet explains how this process is carried out.

Indices or Powers

This is a complete workbook on Indices covering definitions, rules and lots of examples and exercises.
It can be used as a free-standing resource, or can be read in conjunction with mathtutor - the companion on-disk resource.

Linear equations in one variable

This is a complete workbook introducing the solution of a single linear equation in one variable. It contains plenty of examples and exercises.
It can be used as a free-standing resource or in conjunction with the mathtutor DVD.

Logarithms

This booklet explains what is meant by a logarithm. It states and illustrates the laws of llogarithms. It explains the standard bases 10 and e.
Finally it shows how logarithms can be used to solve certain types of equations.

Mathematical language

This introductory booklet describes conventions used in mathematical work and gives information on the appropriate use of symbols.

Polynomial division

Polynomial division is a process used to simplify certain sorts of algebraic fraction. It is very similar to long division of numbers. This booklet describes how the process is carried out.

Quadratic equations

This booklet explains how quadratic equations can be solved by factorisation, by completing the square, using a formula, and by
drawing graphs.

Simplifying Fractions

This booklet explains how an algebraic fraction can be expressed in its lowest terms, or simplest form.

Simultaneous linear equations

This is a complete workbook introducing the solution of a pair of simultaneous linear equations. It contains plenty of examples and exercises.
It can be used as a free-standing resource or in conjunction with the mathtutor DVD.

Solving inequalities

This booklet explains linear and quadratic inequalities and how they can be solved algebraically and graphically.
It includes information on inequalities in which the modulus symbol is used.

Substitution and formulae

Formulae are used to relate physical quantities to each other. They provide rules so that if we know the values of certain quantities we can calculate the values of others. This booklet discusses several formulae.

Transposition, or rearranging formulae

It is often necessary to rearrange a formula in order to write it in a different, yet equivalent form. This booklet explains how this is done.

### Test Yourself (35)

Diagnostic Test - Brackets

Diagnostic test for using brackets.
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Diagnostic Test - Completing the square 1

Diagnostic test for completing the square. (1 of 2).
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Diagnostic Test - Completing the square 2

Diagnostic test for completing the square. (2 of 2).
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Diagnostic Test - Cubics

Diagnostic test for cubic equations.
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Diagnostic Test - Factorising quadratics 1

Diagnostic test for factorising quadratics. (1 of 2).
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Diagnostic Test - Factorising quadratics 2

Diagnostic test for factorising quadratics. (2 of 2).
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Diagnostic Test - Inequalities

Diagnostic test for solving inequalities.
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Diagnostic Test - Linear equations

Diagnostic test for linear equations.
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Diagnostic Test - Logarithms

Diagnostic test for logs and logarithms.
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Diagnostic Test - Mathematical language

Diagnostic test for mathematical language
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Diagnostic Test - Partial fractions

Diagnostic test for partial fractions.
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Diagnostic Test - Pascal's Triangle and the binomial expansion

This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Diagnostic Test - Polynomial division

Diagnostic test for polynomial division.
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Diagnostic Test - Powers

Diagnostic test for powers and indices.
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Diagnostic Test - Simplifying fractions

Diagnostic test for simplifying fractions.
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Diagnostic Test - Simultaneous equations

Diagnostic test for simultaneous equations.
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Diagnostic Test - Substitution

Diagnostic test for substituting formulae.
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Diagnostic Test - Transposing formulae

Diagnostic test for transposing formulae.
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Exercise - Brackets

Online exercise on using brackets.
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Exercise - Completing the square

Online exercise on completing the square.
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Exercise - Completing the square - maxima and minima

Online exercise on completing the square using maxima and minima.
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Exercise - Cubics

Online exercise on cubic equations.
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Exercise - Factorising Quadratics 1

Online exercise on factorising quadratics.
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Exercise - Factorising Quadratics 2

Online exercise on factorising quadratics (2).
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Exercise - Fractions

Online exercise on partial fractions.
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Exercise - Inequalities

Online exercise on inequalities.
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Exercise - Linear equations

Online exercise on linear equations.
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Exercise - Logarithms

Online exercise for logarithms.
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Exercise - Partial fractions

Online exercise on partial fractions.
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Exercise - Pascal's Triangle and the binomial expansion

This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Exercise - Polynomial division

Online exercise on polynomial division.
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Exercise - Powers

Online exercise for powers.
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Exercise - Simultaneous equations

Online exercise on simultaneous equations.
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Exercise - Substitution

Online exercise on substitution.
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Exercise - Transposing formulae

Online exercise on transposing formulae.
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

### Video (17)

Completing the Square - by Inspection

In this unit we consider how quadratic expressions can be written in an equivalent form using the technique known as completing the square. This technique has applications in a number of areas, but we will see an example of its use in solving a quadratic equation.
(Mathtutor Video Tutorial)
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Completing the Square - maxima & maxima

Completing the square is an algebraic technique which has several applications. These include the solution of quadratic equations. In this unit we use it to find the maximum or minimum values of quadratic functions.
(Mathtutor Video Tutorial)
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Expanding & Removing Brackets

In this unit we see how to expand an expression containing brackets. By this we mean to rewrite the expression in an equivalent form without any brackets in. Fluency with this sort of algebraic manipulation is an essential skill which is vital for further study.
(Mathtutor Video Tutorial)
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Factorising Quadratic Equations

An essential skill in many applications is the ability to factorise quadratic expressions. In this unit you will see that this can be thought of as reversing the process used to 'remove' or 'multiply-out' brackets from an expression.
(Mathtutor Video Tutorial)
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Logarithms

Logarithms appear in all sorts of calculations in engineering and science, business and economics. Before the days of calculators they were used to assist in the process of multiplication by replacing the operation of multiplication by addition. Similarly, they enabled the operation of division to be replaced by subtraction. They remain important in other ways, one of which is that they provide the underlying theory of the logarithm function. This has applications in many fields, for example, the decibel scale in acoustics.
(Mathtutor Video Tutorial)
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Mathematical language

This introductory section provides useful background material on the importance of symbols in mathematical work. It describes conventions used by mathematicians, engineers, and scientists.
(Mathtutor Video Tutorial)
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Partial Fractions

After viewing this tutorial, you should be able to explain the meaning of the terms 'proper fraction' and 'improper fraction', and express an algebraic fraction as the sum of its partial fractions. (Mathtutor Video Tutorial) algebraic fraction as the sum of its partial fractions.
(Mathtutor Video Tutorial)
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Polynomial Division

In order to simplify certain sorts of algebraic fraction we need a process known as polynomial division. This unit describes this process.
(Mathtutor Video Tutorial)
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Powers

A knowledge of powers, or indices as they are often called, is essential for an understanding of most algebraic processes. In this section you will learn about powers and rules for manipulating them through a number of worked examples.
(Mathtutor Video Tutorial)
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Simple Linear Equations

In this unit we give examples of simple linear equations and show you how these can be solved. In any equation there is an unknown quantity, x say, that we are trying to find. In a linear equation this unknown quantity will appear only as a multiple of x, and not as a function of x such as x

^{2}, x^{3}, sin x and so on. Linear equations occur so frequently in the solution of other problems that a thorough understanding of them is essential. (Mathtutor Video Tutorial) This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.
Simplifying Algebraic Fractions

This video explains how algebraic fractions can be simplified by cancelling common factors. (Mathtutor Video Tutorial)
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Simultaneous Linear Equations Part 1

The purpose of this section is to look at the solution of simultaneous linear equations. We will see that solving a pair of simultaneous equations is equivalent to finding the location of the point of intersection of two straight lines.
(Mathtutor Video Tutorial)
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Solving Cubic Equations

All cubic equations have either one real root, or three real roots. In this video we explore why this is so. (Mathtutor Video Tutorial)
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Solving Inequalities

This video explains linear and quadratic inequalities and how they can be solved algebraically and graphically. It includes information on inequalities in which the modulus symbol is used. (Mathtutor Video Tutorial)
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Solving Quadratic Equations

This unit is about the solution of quadratic equations. These take the form ax

^{2}+bx+c = 0. We will look at four methods: solution by factorisation, solution by completing the square, solution using a formula, and solution using graphs. (Mathtutor Video Tutorial) This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.
Substitution & Formulae

In mathematics, engineering and science, formulae are used to relate physical quantities to each other. They provide rules so that if we know the values of certain quantities; we can calculate the values of others. In this video we discuss several formulae and illustrate how they are used.
(Mathtutor Video Tutorial)
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.

Transposition or Re-arranging Formulae

It is often useful to rearrange, or transpose, a formula in order to write it in a different, but equivalent form. This unit explains the procedure for doing this.
(Mathtutor Video Tutorial)
This resource is released under a Creative Commons license Attribution-Non-Commercial-No Derivative Works and the copyright is held by Skillbank Solutions Ltd.