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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:



Factorisation

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.

Solution

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
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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: