# A-level Mathematics/CIE/Pure Mathematics 1/Quadratics

## Completing the Square

edit**Completing the square** is a method of converting an expression of the form into an expression of the form .

It relies on the fact that .

First, we take the expression and factor out to get .

Then, we need to recognise that

### Examples

editSuppose we need to convert to completed-square form.

Here, is 1, so we don't need to do anything to factor it out.

Next, we recognise that , and need to find this in the expression.

Thus, we have our answer:

## The Discriminant

editThe **discriminant** is a value that we can use to determine how many real roots the quadratic function has. A *real root* is where the value of a quadratic expression is equal to zero.

The discriminant for an expression is calculated .

If the discriminant is greater than zero, there are two separate real roots.

If the discriminant is equal to zero, there is one *reapeated root*.

If the discriminant is less than zero, there are no real roots.

## Solving Quadratics

editThere are three main methods for solving a quadratic equation or inequality: *factorising*, *completing the square*, and using the *quadratic formula*.

### Factorising

edit**Factorisation** is where we break the expression into its factors.

e.g. can be factorised as

Factorisation can be used to solve equations: if the product of two factors is equal to zero, that means that one of the factors has to be equal to zero.

e.g. Solve

To factorise an expression with a coefficient attached to the term, simply divide out the coefficient

e.g. Solve

However, not all expressions can be factorised.

### Completing the Square

edit**Completing the square** is where we convert a quadratic equation from the form to the form . This makes it easier to solve equations, and it works in all cases, unlike factorisation.

e.g. Solve

### Quardatic Formula

editThe **quadratic formula** states that:

e.g. Solve

You may have noticed that the part under the square root is the discriminant. The reason this makes sense is that if the discriminant is negative, the square root cannot result in a real number, and thus there are no real roots. If the discriminant is zero, then , so there is one repeated root. This leaves the case of when the discriminant is positive, resulting in two real roots.

## Solving Simultaneous Equations

editSometimes we will need to solve simultaneous equations which involve both a linear and a quadratic equation. To solve them, we need to use the method of substitution.

e.g. Solve the simultaneous equations and

## Recognising Quadratics

editSometimes quadratics will be hidden in other forms. If you can make a substitution to turn an expression into a quadratic, you can then solve it as you would a quadratic.

e.g. Find the value of x in