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Factorisation can be seen as the reverse of calculating the product of factors. In order to factorise a quadratic, we need to find the factors which when multiplied together equal the original quadratic.
Let us consider a quadratic that is of the form
Another type of quadratic is made up of the difference of squares. We know that:
This is true for any values of and , and more importantly since it is an equality, we can also write:
This means that if we ever come across a quadratic that is made up of a difference of squares, we can immediately write down what the factors are.
Find the factors of .
We see that the quadratic is a difference of squares because:
and
The factors of
These types of quadratics are very simple to factorise. However, many quadratics do not fall into these categories and we need a more general method to factorise quadratics like
We can learn about how to factorise quadratics by looking at how two binomials are multiplied to get a quadratic. For example, is multiplied out as:
We see that the
So, how do we use this information to factorise the quadratic?
Let us start with factorising
Next, decide upon the factors of 6. Since the 6 is positive, these are:
Factors of 6 | |
1 | 6 |
2 | 3 |
-1 | -6 |
-2 | -3 |
Therefore, we have four possibilities:
Option 1 | Option 2 | Option 3 | Option 4 |
Next, we expand each set of brackets to see which option gives us the correct middle term.
Option 1 | Option 2 | Option 3 | Option 4 |
We see that Option 3 (x+2)(x+3) is the correct solution. As you have seen that the process of factorising a quadratic is mostly trial and error, there is some information that can be used to simplify the process.
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