Factorising Quadratic Equations

An important point to make clear on this subject is as follows:

Not all quadratic equations can be factorised!

In general, the general form of a quadratic equation is ax² + bx + c where x is the unknown variable and a, b, and c are constants. For a quadratic equation to be factorised, there must exist a pair of numbers which add to make b, and which multiply to make the product of a and c. This is expressed below for a pair of numbers, p and q:

p + q = b and pq=ac

Most quadratic equations in GCSE exams have a=1 so we can simplify this in some cases by saying that pq=c. Once we know the values of p and q, we factorise the equation by making it look something like (x+p)(x+q). Let`s demonstrate this with an example below:

______________ EXAMPLE ____________________________-

Factorise x² + 5x + 6

We need to find two numbers which add to make 5 and which multiply to make 6. We can start by looking at 1 and 4. These two numbers add to 5, but they multiply to make 4, when we want 6. Let`s try again. 2 and 3 add to make 5, and they multiply to make 6. This means 2 and 3 are the numbers we were after. We can now say that:

x² + 5x + 6 = (x+2)(x+3)

We can check if this is correct by expanding out what we`ve just done to check that it is in fact equal to x² + 5x + 6. Expanding gives us (x)(x) + (x)(3) + (2)(x) + (2)(3) = x² + 3x + 2x + 6 = x² + 5x + 6. This is correct! We have successfully factorised the above equation into (x+2)(x+3).