Using a grid can be a helpful way to factor a trinomial. The objective of factoring a trinomial is to begin with an expression such as
![\[ax^2+bx+c\]](https://tentotwelvemath.com/wp-content/ql-cache/quicklatex.com-9634679141000cbaa21856fb65f18382_l3.png "Rendered by QuickLaTeX.com")
where 
and 
have no common factors*; and write an equivalent expression
![\[ax^2+bx+c=(mx+r)(nx+t)\]](https://tentotwelvemath.com/wp-content/ql-cache/quicklatex.com-6d5fa73fd9019ff74e949472eaa10cfe_l3.png "Rendered by QuickLaTeX.com")
For example
![\[2x^2+5x-12=(2x-3)(x+4)\]](https://tentotwelvemath.com/wp-content/ql-cache/quicklatex.com-0e48840aa8dbbf0ef7c97c7477584f33_l3.png "Rendered by QuickLaTeX.com")
The grid method can also be used with an unreduced quadratic. See ‘method 2’ below.
Highest Common Factor
In order to use the grid method, we need to be able to take the highest common factor of two terms.
For example, the highest common factor of 
and 
is 
.
The highest common factor of 
and 
is 
.
We can write this using shorthand notation:
HCF
HCF
Review highest common factor here.
The Grid Method
In general, the grid is filled out as follows:
Let’s examine 
We begin by drawing a 2 by 2 grid, with space on top and on the left as follows:
We first enter the 
term and the term as follows.
We then need to find two numbers 
and 
such that
![\[p \times q = a \times c \text{ and } p + q = b\]](https://tentotwelvemath.com/wp-content/ql-cache/quicklatex.com-ce111db581e7d34fe3e62cd0bb6e6b00_l3.png "Rendered by QuickLaTeX.com")
In our example, that is
![\[p \times q = 2 \times -12=-24 \text{ and } p + q = 5\]](https://tentotwelvemath.com/wp-content/ql-cache/quicklatex.com-1043b63880529e0ed4d71f426e55eb2e_l3.png "Rendered by QuickLaTeX.com")
Study the factor pairs of 24, remembering that to multiply to a negative, one of the factors must be negative. The list includes: 
. The factor pair 
works using 
:
![\[-3 \times 8= -24 \text{ and } -3 + 8 = 5\]](https://tentotwelvemath.com/wp-content/ql-cache/quicklatex.com-6a8de2721d234da4ceed2f31eaabdb46_l3.png "Rendered by QuickLaTeX.com")
We complete the grid using these two values as follows (either way around):
If the quadratic is reduced (that is, HCF
), we can now simply take the highest common factor from each row/column as follows:
First row:
Second row:
First column:
Second column:
Check that each term in the grid is the product of green terms in the same row and column:
On the left, we see our factor 
and on the top row we see the factor 
. Therefore,
Check by expanding the brackets.
Method 2
If the quadratic is not reduced, we take the highest common factor from any one row or column, and then complete the remaining boxes as follows:
Highest common factor from any one row or column:
What multiplies 
to get 
? Complete first column:
What multiplies 
to get 
? Complete second row:
What multiplies to get 
? Complete second column.
Check the last box works too:
If the quadratic was not already reduced, you will see a common factor in at least one of your two factors.
Practice