Factorising Quadratics (OIOI Method)
Neil Trivedi
Teacher
Contents
Factorising Quadratics (OIOI Method)
Quadratic expressions are algebraic expressions of the form
where and are integers and .
When we factorise a quadratic expression, we rewrite it as a product of two linear factors, written in brackets. Factorising is the reverse of expanding brackets.
Factorising Quadratics where the coefficient of is
First, we will focus on quadratics where the coefficient of is (i.e. ).
These can be factorised by finding two numbers that multiply to give the constant term, represented by , and add to give the coefficient of , represented by .
Suppose we have the quadratic
This will factorise to give two linear expressions.
First, we look at what multiplies to give . Since , we write in both brackets.
Next, we look for two numbers that multiply to give and add to give the coefficient of which, in this case, is . We list the factor pairs of .
The pairs are:
and
and
and
Out of these pairs, only and can make . To get , we do . Therefore, the quadratic factorises to
Note that this can also be written the other way around.
Example 1:
Factorise the following quadratics:
a)
Step 1: Determine what multiplies to give .
First, we look at what multiplies to give . Since , we write in both brackets.
Step 2: Find two numbers that multiply to give the constant term and add to give the coefficient of .
We look for two numbers that multiply to give and add to give . We list the factor pairs of .
The pairs are:
and
and
and
Out of these pairs, only and can make . To get , we do . Therefore, the quadratic factorises to
b)
Step 1: Determine what multiplies to give .
Just like part (a), . So, we write in both brackets.
Step 2: Find two numbers that multiply to give the constant term and add to give the coefficient of .
We look for two numbers that multiply to give and add to give . We list the factor pairs of .
The options are:
and
and
and
Out of these pairs, only and can make . To get , we do . Therefore, the quadratic factorises to
The Difference of Two Squares
The Difference of Two Squares Principle
The difference of two squares occurs when a quadratic expression has no middle term and is written as the subtraction of two square numbers. To see how it works, let’s consider the expression
There is no term written, but we can think of this expression as
This allows us to factorise using the standard quadratic factorising method.
First, we look at what multiplies to give . Since , we write in both brackets.
Next, we look for two numbers that multiply to give and add to give the coefficient of which, in this case, is . We list the factor pairs of .
The pairs are:
and
and
Out of these pairs, only and can make . To get , we do . Therefore, the quadratic factorises to
We can expand the brackets to check how this works.
The middle terms cancel, leaving
This explains why the original expression has no term.
In general, if a quadratic has no middle term and both terms are square numbers, we can use the difference of two squares principle to factorise it.
Overall, when using the difference of two squares, take the square roots of the two square terms and place them in brackets, one with a plus sign and one with a minus sign.
Example 2:
Factorise the following quadratics:
a)
Single Step: Use the difference of two squares principle to factorise the expression.
b)
Single Step: Use the difference of two squares principle to factorise the expression.
c)
Single Step: Use the difference of two squares principle to factorise the expression.
d)
Single Step: Use the difference of two squares principle to factorise the expression.
Note: The order here matters. Since the is second, the needs to second in the bracket.
Factorising Quadratics where the coefficient of is not
We have factorised quadratics where the coefficient of is . In these cases, once we find two numbers that multiply to give the constant term and add to give the coefficient of , the brackets can be written in either order, as both arrangements give the same middle term.
However, when the coefficient of the term is not , the order of the numbers in the brackets matters. To decide which arrangement works, we use a method known as the OI-OI method, coined by NeilDoesMaths.
The OI-OI method comes from expanding brackets. When two brackets are expanded, the middle term is formed by adding the outer and inner products (as remembered by FOIL).
When factorising, we reverse this process and use the outer and inner products to check which arrangement will give the correct middle term.
Although the OI-OI method can be used when the coefficient of is , it is not essential in those cases because the order of the brackets does not affect the middle term.
For example, consider the quadratic
First, we look at what multiplies to give . This is and , so we write these in the brackets.
Next, we look for two numbers that multiply to give . The only factor pair of is and .
We now decide which bracket the and should go in. The options are
or
Then, we choose one of these options and use the OI-OI method to check whether the outer and inner products, when added together, give . Let’s choose the first option.
The outer product is and the inner product is . However, we cannot make out of these so we choose the second option and verify it using OI-OI.
Here, the outer product is and the inner product is . We can make out of this by doing . Therefore, the quadratic factorises to
Example 3:
Factorise the following quadratics:
a)
Step 1: Determine what multiplies to give .
First, we look at what multiplies to give . This is and , so we write these in the brackets.
Step 2: Find two numbers that multiply to give the constant term.
We now look for two numbers that multiply to give . We list the factor pairs of .
The pairs are:
and
and
It may not immediately clear which pair is correct but, generally, we should first choose the pair whose values are closest together, which will be and .
We now decide which bracket the and should go in. The options are
or
Step 3: Use the OI-OI method to find the correct arrangement.
We now apply the OI-OI method to check which arrangement gives the correct middle term, which is . Let's choose the first option.
Here, the outer product is and the inner product is . We can make by doing . Therefore, the quadratic factorises to
b)
Step 1: Determine what multiplies to give .
First, we look at what multiplies to give . We have two options for this.
and
and
In general, we first choose the pair whose values are closest together, which will be and , so we write these in the brackets
Step 2: Find two numbers that multiply to give the constant term.
We now look for two numbers that multiply to give . We list the factor pairs of .
The pairs are:
and
and
and
Since the numbers and are closest together, we try these first. We now decide which bracket the and should go in. The options are
or
However, both options contain a bracket that can be further factorised, namely . We can factorise a from this bracket. The same issue would occur with the other factor pairs, because each pair contains at least one even number. This would lead to at least one bracket containing and an even number, which we can factorise a from. Since the original quadratic is not already factorised, this tells us that choosing and is not suitable. So, we choose and instead.
We can still use the factor pair and . We now have new options
or
We can immediately rule out the second option, as there is a common factor of in the first bracket.
Step 3: Use the OI-OI method to find the correct arrangement.
For the first option, we apply the OI-OI method to check whether this arrangement gives the correct middle term, which is .
Here, the outer product is and the inner product is . We can make by doing . Therefore, the quadratic factorises to
c)
Step 1: Determine what multiplies to give .
First, we look at what multiplies to give . We have three options for this.
and
and
and
In general, we should first choose the pair whose values are closest together, which will be and , so we write these in the brackets
Step 2: Find two numbers that multiply to give the constant term.
We now look for two numbers that multiply to give . The only factor pair of is and . We now decide which bracket the and should go in. The options are
or
Step 3: Use the OI-OI method to find the correct arrangement.
We now apply the OI-OI method to check which arrangement gives the correct middle term, which is . Let's choose the first option.
Here, the outer product is and the inner product is . We cannot make out of these so we choose the second option and verify it using OI-OI.
Here, the outer product is and the inner product is . We can make by doing . Therefore, the quadratic factorises to
Practice Questions