# How to Solve Quadratic Equations Using Factorisation (Mathematics Lesson)

# What Is Factorisation?

Factorisation is a way of simplifying a quadratic equation into two brackets:It is the opposite of expanding the two brackets out using the FOIL method.

# How Factorisation Solves Quadratic Equations

Solving a quadratic equation involves finding two values of**x**that make the equation = 0.

Since equating either factor to 0 makes their product 0, two equations can be solved to find the two roots:

# How to Factorize and Solve Quadratic Equations

Consider factorizing only quadratic equations where the of the x^{2}term,

**a**= 1. :

**Question**: Factorize and solve the quadratic equation below:

**Factorisation**

The aim is to factorize :

Step 1

**a**,

**b**and

**c**.

**a**= 1,

**b**= 5,

**c**= 4.

Step 2

**c**term is the product of the roots.

**c = x**.

_{1}x_{2}List the pairs of factors that multiply to make the

**c**term:

Step 3

**b**term is the sum of the roots.

**b = x**.

_{1}+ x_{2}Find the pair of factors found in

**Step 2**that add up to the

**b**term:

Step 4

x

_{1}= 1, x

_{2}= 4.

**Note**: Use the FOIL method to check that if these brackets are expanded, the original quadratic equation is obtained.

**Solution**

Step 5

**x**:

_{1}Step 6

**x**:

_{2}The solution to the quadratic equation x

^{2}+ 5x + 4 = 0 is

**x = -1**or

**x = -4**.

# A Real Example of How to Factorize and Solve Quadratic Equations

An additional complication when factorizing is that the in the quadratic equation,**b**and

**c**, may have minus signs.

The slider below shows how to factorize a quadratic equation when some of the are negative.

##### Interactive Test

**show**

##### Note

# FACTORISING AND FACTORING

The terms factorizing, factorizing and factoring are sometimes used for this method, depending on country.**WHAT'S IN A NAME?**

The word 'quadratic' comes from the word 'quad', meaning "square" - because the

**x**is squared.

##### Note

# WHY FACTORISATION WORKS

Factorisation is the opposite of expanding two brackets using the FOIL method.Let's go backwards. Start with the quadratic equation factorized into two brackets:

Expand the brackets using the FOIL method:

Compare this with the standard form of a quadratic equation (with

**a = 1**):

By comparing co-efficients, it is easily seen that:

# BE CAREFUL WITH SIGNS 1

When a quadratic equation has been factorized, the roots of the equation can be read off.But remember, you need to flip the sign. For instance, if the factorized equation is:

The roots are:

Another example, if the factorized equation is:

The roots are:

Finally, if the factorized equation is:

The roots are:

# BE CAREFUL WITH SIGNS 2

Depending on the sign of the**b**and

**c**terms, the numbers you write it the bracket can be positive or negative.

The picture below defines what is meant by a positive or negative

**b**,

**c**and number in a bracket:

The following table gives a quick summary of what the signs must be: