Factorising quadratics without trial and improvement.

How do you teach GCSE students how to factorise an expression like;


I was taught to solve through a very time consuming trial and improvement method with no reason given as to how it and why it worked. Since those days I have found much better ways to solve these problems. Take the example above. My favourite and very satisfying approach is to multiply the constant by the coefficient of x², find a pair of factors of this product that sum to the coefficient of x, replace the coefficient of x with the sum of these chosen factors, organise the expression into two brackets and factorise this to get to the solution. What a sentence to absorb! broken down it looks like this…




For those of you who are familiar with my blogs you know that what I have just done is give you a method with no reason as to why it has worked. You will also know that I am more interested in why it works. We all agree that this is what all good teachers do by the way, they show pupils how and why an idea works.


In general a quadratic can be written ax² + bx + c where a, b and c are constants where a ≠ 0. However I think this is unhelpful with respect to what we are trying to teach. I much prefer to use acx² + adx + bcx + bd. Now, I agree this looks much more complicated but if we place each element into a 2 x 2 grid we have the following.

Without too much detective work it is possible to discover what has been multiplied in order for the cells of the grid to be filled in.


The grid tells us that (ax + b)(cx + d) = acx² + bcx + adx + bd. The right hand side can be partially factorised at this stage so that acx² + bcx + adx + bd = acx² + x(bc + ad) + bd.

You may have noticed already that the leading left diagonal contains a, b, c and d. 

You may even have concluded that the factors of the sum of the coefficients of x are found by multiplying the factors of and the constant.


This is worth presenting to learners. This is why the multiplication of the coefficient of x² and the constant helps us to use mathematical reasoning as opposed to long winded trial and improvement. We have looked at a general quadratic expression and because of the pure nature of the algebra, the method is watertight!

So finally, we will test this out on the specific expression 12x² – 17x + 6.


Factorise these for yourself. Are you convinced?

a)  2x² + x + 1    b) 2x² – x – 6    c) 6a² – 11a + 3

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