Many students who are comfortable factorising quadratics of the form x^{2} + bx + c struggle when there is a coefficient in front of the x term, as in ax^{2} + bx + c. They usually understand what is required, but become frustrated through the apparent ‘trial and error’ process that they apply until they identify the correct solution.

When I was a student at school, I remember being taught a structured trial and error process that worked methodically through all the possible combinations. It worked every time, but it was laborious and the greater the number of factors that the ‘a’ and ‘c’ terms had, the more tiresome the process became.

As a teacher, I continually look for improved methods of carrying out this process. However, many of them lose the understanding of what is going on – and students can end up struggling to remember a method, rather than remember what they are trying to achieve. To this end, I prefer for students to begin with a trial and error method, so that they appreciate the nature of the problem, before moving onto an appropriate method.

Consider the example 2x^{2} – x – 6 in order to study some of those methods.

Most methods start by multiplying the ‘a’ and ‘c’ terms…

…then finding factor pairs of their product, i.e. of -12…

…and identifying the pair that sums to give the ‘b’ term (i.e. -1). In the chosen example, this gives 3 and -4.

The ‘Lizzie method’ separates the pair, as shown, divides by the ‘a’ term, then multiplies the ‘x’ and the fraction (simplifying if necessary) until the denominator has been multiplied out.

A variation (see here for more detail) on the Lizzie method, separates the pair and multiplies the ‘x’ by the ‘a’ term, dividing by any common factors.

However, in my mind, these are rather formulaic and I prefer a method which is more intuitive. The method that I like to use (see here for more detail) involves rewriting the ‘b’ term as the sum of the factor pairs. The expression is then factorised in two parts; there is no need to have to ‘remember’ which number to write where, and which terms to multiply or divide by other terms.

OrHere is another example, 6x^{2} – 5x – 6.

Multiply ‘a’ and ‘c’ terms | Find factor pairs of -36 | First method |

Second method | Third method | Third method, again |

However, as a teacher, judgement is required as to which method, if any, may be more appropriate for the class, or just for individuals within it, to use. Experience has shown that time spent explaining lots of methods to lots of students can be detrimental. Many students don’t want to see more than one method, so finding time (*sigh*) to individualise explanations would be beneficial – even if this is done ‘out of class’, or otherwise.