Quickest way to factor quadratic equations

Question

hi guys, in one of the OG questions i found this quadratic equation

x^2 + 12x - 540

I am looking for a quick method to factorize it.

TestPrepUnlimited · Answer

Let's start from anywhere to break down 540, for example,  540 = 20 * 27 . We are looking for a difference of 12, but the difference of 20 and 27 is only 7 so we would like to expand this difference. We need a different product that results in 540, with some factor slightly bigger than 27. The next factor of 540 we can try is 30, so 540 = 30*18, which is bingo. Then we have (x + 30)*(x - 18). A hint is to always start from the 10s, 10 20 and 30 are your best bet.

IanStewart · Answer

The standard method is to look at factors of 540 until you find a pair of factors that give the correct sum. With 540, though, then unless you get lucky with your first guess, that's never going to be lightning-quick, because there are many options. But this is the option you should rely on most of the time.

If you're good with divisibility, there can be shortcuts. Here, we want two numbers that are 12 apart. If they're 12 apart, they're both even or both odd, but if they multiply to 540 they both must be even. And if we get 540, a multiple of 5, as our product, one of our numbers must be an even multiple of 5. So one of the two numbers we're looking for is a multiple of 10 for sure. You might then try 20 and 32, but that product is slightly too big (and 20 and 8 gives a product far too small), and then the correct numbers, 30 and 18, would very likely be the next pair you'd check. Finally, making sure we get a sum of +12 and not -12, we get the factorization (x + 30)(x - 18).

When the number at the end has a lot of divisors, as 540 does, there really isn't a super-fast way to factor. Often however, at least on the GMAT, there will be a way to solve the problem where you can avoid factoring - the GMAT really doesn't expect you to be a human calculator, and builds in shortcuts in problems that can be solved in long algebraic ways. That's the case in the OG question you mention, which is a geometry question if it's the question I think it is (Q134 in QR 2017 PS). In that question, other techniques can be used to help zero in on the solution. Estimation, for example, can let you get the answer without factoring -- if you first guess that line QT is 12, the overall area of the rectangle is far too small, and if you guess it's 20, the overall area is too big. You then know the answer is between 144 and 240, and only one answer choice is possible.

bumpbot · Answer

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Quickest way to factor quadratic equations

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