When factoring quadratic expressions $ax^2+bx+c$ it is common to the guess and check factors (AKA the cross method).

This would involve factoring $a$ and $c$ and considering particular combinations of these factors that add to $b$ taking into account the signs of $a,b,c$. This is typically presented using a cross and just the coefficients.

I am worried about this in that there seems to be no way to record what guesses have been tried, and no way to record the "checksums". See Quadratic Equations on wikipedia. (Surely this search is justified by Gauss's Lemma and won't work if the quadratic is irreducible.)

I would like a way to explicitly and concisely enumerate all possible combinations. From this, one could demonstrate heuristics to simplify the process.

Let's take the monic $x^2+5x+6$ with all positive coefficients.

One way would be to systematically list of all factors which give the correct leading coefficient and constant and chose one which gives the correct cross term. $$ (x+1)(x+6)\\ (x+2)(x+3)\\ (x+3)(x+2)\\ (x+6)(x+1) $$ From this we see the middle two give the right answer.

I have tried something like:


where the rows and columns are labelled by the factors of $c$ and the cells contain the sums. I'm unsure how this extends to negative coefficients and non-monic quadratics.

  • 1
    $\begingroup$ Could you briefly summarize what you refer to as the "cross method"? I've never encountered that terminology. $\endgroup$
    – amWhy
    May 1 '19 at 22:15
  • $\begingroup$ To @Namaste 's question for clarification, as I've never heard the term cross method, even though I have a cross, it strikes me that there are methods of teaching that are not "official" but have pet names assigned. My own answer below - I learned from a student, the teachers that use this call it either A-C or Split the Middle, and others, use pure guess/check for all possible numbers. $\endgroup$ May 1 '19 at 22:51

For factoring a quadratic, when "A" = 1, it's relatively easy. Answer the question, "What 2 numbers multiply to C but add to B?" This process may be guess and check, but when A is 1, there's little need to spend much time on this. When one factor has a negative integer, the rule doesn't change, so much as the student needs to be mindful of where the minus sign lands.

It's when A is non-zero we need to spend more time. The A-C Method, aka "Split the middle term" is useful in this case. This is my notes for that -

enter image description here

Above, to solve for the 2 numbers AC=60, B=19, I have students start listing the pairs of number that multiply to 60, along with their sum -

  • 1 + 60 = 61
  • 2 + 30 = 32
  • 3 + 20 = 23
  • 4 + 15 = 19 (Done)

And an example when there's negative numbers.

enter image description here

Now we needed 2 numbers, adding to -1, but multiplying to -72. We can use the same method by looking for the pairs that multiply to 72 but have a difference of 1.

  • 36 - 2 = 34
  • 24 - 3 = 21
  • 18 - 4 = 14
  • 12 - 6 = 6
  • 9 - 8 = 1 >> 9 & 8 have a difference of 1, but since we want the sum to be -1, we use -9 and 8.

Note: The tutorial above assumes the student is already proficient at factoring when A = 1, so these extra details aren't on these sheets, but would appear on the lesson for the other cases.

  • $\begingroup$ See also the A-C Method from Bill Dubuque. $\endgroup$ May 1 '19 at 22:55
  • $\begingroup$ @JoeTaxpayer The bit I'm interested in is between "We need 2 numbers..." and "It's 15 and 4." How does one enumerate the possible combinations? Also, this example doesn't involve negative numbers. $\endgroup$
    – pdmclean
    May 2 '19 at 4:33
  • $\begingroup$ @pdmclean - I've updated to (hopefully) address both issues you cited. In hindsight, including the extra details would have made this tutorial more comprehensive, and a helpful reminder to the student who misplaced or forgot the first. $\endgroup$ May 2 '19 at 9:28

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