When factoring a quadratic expression using the grid method, we assume that and do not share any common factors (other than 1), and we assume that our expression does indeed factor, that is, there exist such that

Given the values and , how does the grid method correctly calculate and ?

Since HCF, neither binomial will have a common factor: HCF and HCF.

By expanding the right hand side, we have

It is immediately apparent that and , however this does not help us calculate the values .

We also notice when we multiply two binomials of this kind that there are two terms.

Let us break the our given value of into two values, and , where :

# Calculating p, q given a, b, c

Let’s put this information into the grid format.

Given that we have:

Given that we have:

In the second version, notice that when we multiply the two diagonals, we have the same expression:

Therefore, from the first grid we have

Which leads us to conclude .

We already know that (that is how we defined and ).

This leads us to derive values for and directly from the coefficients and :

*If there are no integer solutions for and , then the values m, n, r, t are not integers. Our quadratic expression does not factor with integers.*

By solving this system to calculate and we can fill in the two by two grid:

# Calculating m, n, r, t

At this point in the process, values and are unknown, however the values in the grid are complete. In terms of , the two by two grid is:

Let’s consider the first row (any row or column will do)

Notice that is common to both terms in this row. Also, as we noted in the beginning, HCF, therefore is actually the highest common factor in this row.

Now let’s consider the first column. By a similar argument, we notice that is the HCF of this column (because HCF). We also notice that we need to multiply with to get the required product :

We continue in this way, drawing out the required factor in each row and column:

Beginning with the values and , we have calculated and .

The example once again:

This grid above shows the values and . By drawing out the HCF of each row/column, we get and :

Here, ; , and .