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Fast way to factor?

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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Fast way to factor?

by gmatfelix » Wed Mar 27, 2013 1:24 am
Hey,

I am looking for a more efficient way to factor quadratic equations.
It takes me much too long solving them.

Here an example:
t² + 4t - 672
==> (t - 24)(t + 28)

Rule: x² + px + q
P = a + b
q = a * b
(x + a)(x + b)

My approach is to divide the 672 into factors (2*2*2*2*2*3*7) and then search for pairs that also fit the p=a+b equation. In this case (2*2*2*3=24;2*7*2=28) ==> P=-24+28 ==> a=-24 and b=28

But that way is neither systematic nor fast.
Please give me a hint how to speed this up :)

Thank you!
Felix
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by Anju@Gurome » Wed Mar 27, 2013 1:48 am
What you are doing is the standard and only approach.

However, we can apply some common sense and reduce the search time for the pairs.
Let me explain with your example. We know that product of the pairs should be 672 and sum should be 4. What does this tell us? As the sum is 4, the factors must be close and √672 ≈ √676 = 26 must lie almost midway between them. (Note : The underlined part is only valid when the pair is very close)

Now, our first guess should be (26 - 2) = 24 and (26 + 2) = 28.
Check if they are products of 672 or not.


Let us take another example : t² + 221t - 672
Now, we know that product of the pair should be 672 and sum should be 221. What does this tell us? As the sum is 221, the factors must be far apart from each other. Now, we can pick factors from each extremes and check (like 1*672, 2*336, 2*224, etc) or we can notice that 221 ≈ 672/3, hence, the pair must be 3*(672/3) = 3*224 (Note : Our previous logic, i.e. 26 must lie almost midway doesn't hold here as the factors are far apart.)

Hope that helps.
Anju Agarwal
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by Brian@VeritasPrep » Wed Mar 27, 2013 8:40 am
Great method, Anju!

And one more thought for you on this, Felix - on the GMAT you're never doing algebra in a vacuum...there's always either answer choices or a Data Sufficiency question stem and they'll often give you clues. For example, in this question:

If it's problem solving, it might well ask "which of the following is a possible value of t?"

(A) -24
(B) 14
(C) 4
(D) 24
(E) 28

And you could then think in terms of number properties (the digits of 672 add to a multiple of 3, so -24 or 24 are looking pretty good as first numbers to try factoring since you know that one of the two roots of this problem is a multiple of 3) or gaming the answer choices (since there's both a -24 and a +24, 24 is likely a factor...OR...672 is a pretty big number so it's most likely that 4 isn't a factor, but something in the 20s is).


Or, if it's a Data Sufficiency question, it might ask:

Is t > 25?

In which case you'd want to test numbers in that 22-30 range since you know that one of the factors will be close.


So keep an eye on the structure of the question around the quadratic and you'll often find clues that help you streamline which options you try, and again like Anju said if that middle term is small her method is fantastic.
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by misterholmes » Sat Apr 06, 2013 9:18 am
Dear Felix,

Your approach is perfect in its own place, but GMAT models get scaled up over time, and as the complexity goes up you may need to adjust or even abandon techniques that have worked well before. In particular, you need to build-in a recognition function: Yes, I kind of know this problem, but my solution will take forever here, so what else do I see?

Kind Regards,

Misterholmes
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