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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 x2 term, a = 1. :

Question: Factorize and solve the quadratic equation below:


The aim is to factorize :

Step 1
Find the values of a, b and c.
a = 1, b = 5, c = 4.

Step 2
The c term is the product of the roots. c = x1x2.

List the pairs of factors that multiply to make the c term:

Step 3
The b term is the sum of the roots. b = x1 + x2.

Find the pair of factors found in Step 2 that add up to the b term:

Step 4
Insert these values in the pair of brackets.
x1 = 1, x2 = 4.

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


Step 5
Equate the first bracket to 0, and use algebra to find x1:

Step 6
Equate the second bracket to 0, and use algebra to find x2:

The solution to the quadratic equation x2 + 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


The terms factorizing, factorizing and factoring are sometimes used for this method, depending on country.


The word 'quadratic' comes from the word 'quad', meaning "square" - because the x is squared.



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:


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:


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: