Samwong wrote:
I had problem factoring x^2 + 12x - 540 when I first saw this problem. I now know what factors will work after 5 minutes. I broke 540 into primes: 2^2, 3^4, and 5, but it was still not obivous what number would work. Then I tried to listed all the factor pairs, but that method also took a long time. Can someone please suggest any method that can be used to factor a complicated quadratic equation? Also, what is a efficient method to list all the factor pairs. Thanks.
Posted from my mobile device Dear
Samwong,
I'm happy to help.
The GMAT is not going to ask you to do factor of that level. That, I believe, is beyond the arithmetic they would expect, except on perhaps a very hard 700+ level question.
Yes, that's a hard problem. Here's what I would do if I wanted to factor this quickly. Product = -540 = (2^2)(3^3)(5) = 2*2*3*3*3*5, and sum = +12. Finding the prime factorization of 540 is extremely helpful!
Now, I know if the factors were far apart (e.g. 1 and 540), then there's no way the factors would have a difference of 12. Given the size of the product, 540, the difference is awfully small. If we start checking factors at the smallest factors {1, 2, 3, 4, ...} then we will be taking the long way. Let's start where the factors are closest together.
The factors of a number are closest together around the number's square-root (if the number is a perfect square, then the closest of all factors are the two factors of its square root --- e.g. 8 and 8 are factors of 64). We don't need to know the exact square root of 540 (which would be some ugly decimal) --- we'll just estimate in broad strokes 20^2 = 400 and 30^2 = 900, so the square root of 540 must be somewhere between 20 and 30. That's where we'll start checking.
27 = 3*3*3 could be a factor ---- 540 =
2*2*3*3*3*5, so 27*20 --- too close --- we have to move out from there --- from 20, go down to 18, the next possible factor ---18 = 2*3*3, so 540 =
2*2*3*3*3*5, so 18*30 --- BINGO! That's it. That's a relatively fast way to find the right pair.
Does all this make sense?
Mike
_________________
Mike McGarry
Magoosh Test Prep