Why AC factoring works?

Question

AC factoring can be used to factor quadratic equations. Why does it work?

Answer 1

Answer 2

The AC method, on my opinion, is obsolete. There are several methods that are easier. "Slide and Divide" is one of the variations.

Take 42x² + 47x +10 for example. Multiply the constant and the leading coefficient, which gives you 420. Examine the factor pairs as before until find a pair whose sum is equal to the coefficient of the middle term (47).

So far the effort is the same as before. Now divide the factors by the original leading coefficient:

(12/42) = 2/7: write the first factor as the reduced denominator times x plus (or minus if the factor is negative) the numerator as a constant.

Do the same with the other factor: (35/42) = 5/6

This results in ( 7x + 2 ) * ( 6x + 5). I believe this is much quicker than "factoring by grouping ".

Second example;

8x² + 26x + 15

ac = 8 * 15 = 120

6 * 20 = 120, so use these factors.

6/8 = 3/4 and 20/8 = 5/2

( 4x + 3 ) * ( 2x + 5 )

I think once you've tried it a few times, you will abandon the old ac method.

Note: There are several variations of this: "Slip and Slide" is one version.

Note: Before converting to factored form, you also have the solutions if the quadratic is set equal to zero. An additional bonus.

x = -2/7 and -5/6 in the first case; and

x = -3/4 and x = -5/2 in the second case.

Answer 3

https://youtu.be/LrHp4_YBnYk?si=XX0WptWrKgZTrFy2