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Factoring Quadratics: The Hard Case:
     Examples of How to Use "Box" (page 3 of 4)

Sections: The simple case, The hard case, The weird case

Note that "box" works only if you have first removed all common factors.

• Factor 2x² – 4x – 16

First, I need to remove the common factor of 2 from each term, to get 2x² – 4x – 16 = 2(x² – 2x – 8). Then I need to factor the remaining quadratic: x² – 2x – 8 = (x – 4)(x + 2). I need to be careful not to forget the factored-out "2" when I write down my final answer:

2x² – 4x – 16 = 2(x – 4)(x + 2).

The one special case that often causes students some trouble is when the leading coeffiecient is a negative one. A good first step is to factor out the –1.

• Factor –6x² – x + 2

I will first take out the minus one to get –6x² – x + 2 = –1(6x² + x – 2). (I need to remember that every sign changes when I multiply or divide by a negative. I mustn't fall into the trap of taking the –1 out of only the first term; I must take it out of all three!) Factoring the contents of the parentheses then gives me:

Then:   Copyright © Elizabeth Stapel 1999-2009 All Rights Reserved

–6x² – x + 2 = –1(6x² + x – 2) = –1(2x – 1)(3x + 2)

Putting these two techniques together (factoring out anything common, and taking out a leading negative sign), you can handle such problems as:

• Factor –6x² + 15x + 36

First, I will first remove the common factor of 3, taking the leading negative sign with it: –6x² + 15x + 36 = –3(2x² – 5x – 12). Then I'll factor the remaining quadratic: 2x² – 5x – 12 = (x – 4)(2x + 3):

When I write down my answer, I need to remember to include the –3 factor:

–6x² + 15x + 36 = –3(x – 4)(2x + 3).

A disguised version of this factoring-out-the-negative case is when they give you a backwards quadratic where the squared term is subtracted. For example, if they give you something like 6 + 5x + x², you would just reverse the quadratic to put it back in the "normal" order, and then factor: 6 + 5x + x² = x² + 5x + 6 = (x + 2)(x + 3). You can do this because order doesn't matter in addition. In subtraction, however, order does matter, and you need to be careful with signs.

• Factor 6 + x – x²

First, I will want to reverse the quadratic, but I'll need to take care with the signs: 6 + x – x²
= –x² + x + 6. Then I'll factor out the –1, and factor the remaining quadratic as usual:

–x² + x + 6 = –1(x² – x – 6) = –1(x + 2)(x – 3)

There is one other type of quadratic that looks kind of different, but it works in exactly the same way:

• Factor 6x² + xy – 12y².

This may look bad, what with the y² at the end, but it factors just like all the ones above. Remember, from the "simple" case, that we knew that the factors had to be of the form:

(x + something)(x + something else)

...because we knew we'd multiplied factors that looked like this in order to get the quadratic in the first place. This was how we knew that we needed x's in the fronts of our parentheses. In the same way, we know that we must have multiplied factors of the form:

(an x term + a y term)(another x term + another y term)

...to get that y² term at the end of the quadratic. So we'll need to put y's at the ends of our parentheses. But other than this, the process will work as usual.

First, I need to find factors of (6)(–12) = –72 that add to +1; I'll use +9 and –8.  Then "box" gives me:

So 6x² + xy – 12y² factors as (2x + 3y)(3x – 4y).

Quadratic factoring can pop up in even more exotic forms than the last example above....

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