Anything involving any of the expressions: < less than > greater than ≤ less than or equal to ≥ greater than or equal to is an inequality. There are fundamentally two sorts of inequalities: One sided, or simple inequalities such as 4x-1<3 Two sided inequalities such as Solving inequalities is a bit like solving equations. […]

## Multiplying, Dividing, Adding and Subtracting Fractions and Mixed Fractions

We can multiply, add, subtract or divide fractions. The methods are shown below: Multiplying Fractions– frankly we just multiply: Adding Fractions– We make a common denominator: Subtracting Fractions – Again make a common denominator: Notice the answer is minus. Dividing Fractions: Turn the second fraction upside down and multiply. The reason we do this is […]

## Trigonometry and Right Angled Triangles – Finding a Side

There are three basic formulae, involving the ratio of the lengths of the sides. Given a right angled triangle, we first label the sides for the angle we have: If we have the angle shown, we label the sides for this angle. Opposite =is the side opposite the angle x Hypotenuse =is the longest side, […]

## Compound Interest

If you invest money in a savings account at say 10% per year, typically you are paid compound interest. This means that as soon as the first instalment of interest is paid, you start getting paid interest on your interest. If you were to draw a graph of the amount of money in your account […]

## Factorising Simple Expressions

The simplest expressions to factorise have one common factor and factorise into one bracket. Example: has a common factor of 2. We write down 2 first and then fill out the inside of the bracket so that when it is multiplied out, is the result. Therefore, Sometimes we have expressions with more than one common factor, […]

## Factorising into Two Brackets when the leading term is x squared

We may have to factorise expressions such as When the term is exactly and not for example, this is very simple. We look for two numbers that add to give 7 and multiply to give 12. Two such numbers, the only two in fact are 3 and 4, hence the expression factorises as The order in which we […]

## Completing the Square

Recall the algebraic identity We can use this to complete the square. Consider the quadratic function What can be added to yield a perfect square? Compare the coefficients of : Then Generalizing to any quadratic function of the form , 2e=b which yields e=b/2. Hence Example: Use Complete the Square Method to solve Note that the method […]

## The Sine Rule

We can use the sine rule to 1.Find a side when it is opposite and a known angle and we also have another side opposite another angle, both of which we know: 2. Find an angle when it is opposite an known side and we also have another side opposite another angle, both of which […]

## The Quadratic Formula

A quadratic equation is an equation of the form ax² + bx + c = 0, where a . we can solve equations of this form by identifying a, b and c and substituting them into the formulato find . Example: Solve Then or It is very important to get the correct values of ,, and . For the following equations […]

## Trial and Improvement

Sometimes it happens that we can’t factorise an expression or even use the quadratic formula to solve it, for exampleIf we try and solvewhich is equivalent towe cannot do it by these methods. However if we know that the answer is in a certain range we can keep making educated guesses until we get the […]