Q&A #5391

Factoring quadratics using a box divided in fourths

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From: Pat Ballew (for Teacher2Teacher Service)
Date: Dec 23, 2000 at 20:46:45
Subject: Re: Factoring quadratics using a box divided in fourths

  The box, or gelosia approach is one of many ways teachers try to make
factoring trinomials (and some tetranomials) more reachable by students. I
will do an example step-by-step, and give some general statements as I go
along.  Let's use 8x^2-10x-3 as an example. In the common notation, A=8,
B=-10, and C=-3.
  Notice that I have selected a quadratic that has no common divisor.  If one
exists it should be factored out first. The leading term, 8x^2 goes in the
top left corner of the box, and the final term, -3, goes in the bottom right
I use this placement when I teach it. There are variations.

  Now we need to find the two middle terms.  We need to find two numbers that
multiply to make  A*C (in this problem -24) and add up to make B (in this
case -10)  By sequentially ordering factors of -24 we can find a pair that
sum to -10...
   -24 * 1  = -24  but -24 + 1 = -23
   -12 * 2  = -24  and -12 + 2 = -10  Eureka, we have found it.

The -12 and +2 are the coefficients of the linear terms that have added
together to make -10x, and they go in the two remaining boxes in either

NOTE of EXPLANATION... It is good to illustrate at this point that when two
binomial terms are multiplied using the distributive property (many teachers
use a FOIL name for this method) one of the last steps is to combine two
"like terms" to reduce the four terms to three.  This step we have just used
undoes this combination and reproduces the four terms of the multiplication.

Now we have everything in the box which I have illustrated below:
                    8x^2|   +2x
                    -12x| -3

Now we have two rows and two columns. We find the greatest common factor in
each row to create one binomial factor.  The factor of the top row is 2x,
and the bottom row has a factor of -3, so one of the factors is (2x-3).

Now we do the same for the two columns.  The factor of the left column is 4x,
and for the right we get 1 for the largest factor, so this gives (4x+1) as a

Now we need to check to see that (2x-3)(4x+1) does indeed multiply together
to make 8x^2-10x-3 and we have completed the task.

Some rules about signs are in order.  Expressions factorable with this method
will have an even number of negative terms (zero, two or four).  If the two
negatives terms are both on the same diagonal, then both factors will contain
a negative sign (producing a postive constant term).

I hope this is the method you were searching for, and that my explanation is
of some help to you..

Good luck,
 -Pat Ballew, for the T2T service

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