This is factoring page 3d.   It explains the simplified AC Method

In the first page, I presented the perfect square trinomial and the difference of two squares.

In the second page, I presented a method of factoring four-term expressions with pairs of factors.

Pages 3a, 3b, 3c, and 3d give three methods of factoring trinomials:

Page 3a works by listing the factors of the "a" and "c" coefficients.
Page 3b completes the square.
Page 3c is the "AC Method", which has you multiply the "a" and "c" coefficients, and list all the factors of that product.
Page 3d (this page) is the "Simplified AC Method"

Some people say this is even easier than the AC Method.

How to Factor Using the Simplified AC Method

First, consider the general form of the quadratic expression:

ax² + bx + c

where a, b, and c have no factors in common.

(If they have any factors in common, then "pull out" that factor separately, as in this example:

(2)(ax² + bx + c), if the common factor was 2.

Now look again at EXAMPLE 5:

6x² + 19x - 20

It has no factors in common.  The "a" coefficient is 6, and the "c" coefficient is 20.

Next, consider what happens to this polynomial when we let y=ax, and so x=y/a:

ax² + bx + c
y²/a + by/a + ac/a
(1/a)(y² + by + ac)

This new polynomial is easier to factor, because the high-order coefficient is 1.  To get ac in our example, we multiply the 6 and the -20 together to get -120.  This gives us

(1/6)(y² + 19y - 120)

Next, factor the new polynomial.  In our example, it is

(1/6)(y-5)(y+24)

Now, substitute the value of ax (6x, in our example) in place of y:

(1/6)(6x-5)(6x+24)
(6x-5)(x+4)

Summary

You can factor a quadratic this way:

6x² + 19x - 20
(1/6)(y² + 19y - 120)
(1/6)(y-5)(y+24)
(1/6)(6x-5)(6x+24)
(6x-5)(x+4)

If you weren't able to factor a quadratic equation using any of the methods in this section, then either it can't be factored, or else the factors are not rational numbers.  In either case, you'll find out what's going on using the QUADRATIC FORMULA -- that's the topic of the next factoring page.

Related pages in this website

Next: the quadratic formula

The webmaster and author of this Math Help site is Graeme McRae.
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