Quadratic equations are an integral part of the GCSE Maths curriculum. Yet they can be a real challenge to pupils who have fallen a little behind with algebra. In this brief article, we will go through factorising quadratics step by step to show you how it’s done.
Let us recall that the general form of a quadratic equation is y = ax + bx + c , where a, b and c, are numbers that can be negative, positive or zero. By considering numerical examples we are going to investigate four cases.
1. y = 3x² + 2x
In this case, a = 3, b = 2, c = 0. By collecting the x term we have:
y = x(3x + 2).
However, sometimes we need to collect numbers as well.
For instance, the factorised form of the equation: y = 6x² + 2x is y = 2x(3x + 1).
Remember that you can always check your work by multiplying out of the brackets.
2. y = 25x² + 5
In this case, b = 0 and we can only collect numbers.
The factorised form of the equation is y = 5(5x² + 1).
3. y = x² + 5x + 6
In this case, a = 1, b = 5, c = 6.
Here we need to use a different approach, the first step consists in writing down the following:
+ x
y = (x + ···)(x + ···)
We are looking for two numbers that multiply to 6 and add to 5. We always start working from the right, in this case:
3 x 2 = 6 and 3 + 2 = 5.
Thus, our mysterious numbers are 2 and 3 and the factorised form of our equation is:
y = (x + 2)(x + 3).
Sometimes things are made more complicated by the presence of negative numbers. For instance, let us consider the equation:
y = x² + 2x – 8.
By using the same approach as before, without writing down the signs we have:
+ x
y = (x ···)(x ···).
In this case, 4 x – 2 = – 8 and 4 + – 2 = 2. Thus our numbers are 4 and -2. Hence the factorised form of our equation is:
y = (x – 2)(x + 4).
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Federico Antonelli
Maths tutor at TuitionWorks
I am an applied mathematician and qualified secondary teacher. I have done research in the field of nuclear energy and am currently studying toward a PhD with the University of Cranfield in aerospace materials.
