Advanced Factorization

You will almost never need to use the formula on GMAT. Whenever you have a quadratic that you need to factor, you will almost always get two integral roots (in GMAT). If I am looking at a GMAT question and am unable to find the factors, I will go back and check my quadratic to see if it is correct rather than try to use this formula... I would be that sure of not needing to use it!

When you have something other than 1 as the co-efficient of x^2, you compare the product of the last term and the co-efficient of x^2 with the middle term.
i.e. if the given equation is px^2 + qx + r = 0, you need to find the numbers that multiply to give p*r and add up to give q (when p = 1, you find numbers that multiply to give r (which is 1*r so essentially it is p*r) and add up to give q)

e.g. the question above: 5x^2-34x+24
If you need to factorize it, you need to find two numbers a and b such that:

a + b = -34

a*b = 5*24

Step 1: Prime factorize the product.
a*b = 2*2*2*3*5

Step 2: Check the signs and decide what you need. Here sum is -ve while product is positive. This means a and b both are negative. Both will add to give a negative number and multiply to give a positive number. It also means that a and b both are smaller than 34 (since they will add up to give 34. If one of them were negative and other positive, one number would have been greater than 34. In that case, the product would have been negative.)

Step 3: Try to split the prime numbers into two groups such that their sum is 34. Try the most obvious group first i.e.
2*2*2 and 3*5 ---- 8 and 15 add up to give 23.
But you need 34, i.e. a number greater than 23.

Before we discuss the next step, let me explain one thing:
Let's say we have 2*2*5*5.. I split it into 2 groups 2*5 and 2*5 (10 and 10). Their sum is 20.
I split in in another way 2*2 and 5*5 (4 and 25). Their sum is 29. The sum increased.
I split in in another way 2 and 2*5*5 (2 and 50). Their sum is 52. The sum increased again.

Notice that farther apart the numbers are, the greater is their sum. We get the least sum (20) when the numbers are equal.

Going back to the original question, 8+15 gave us a sum of 23. We need 34 so we need to get the numbers farther from each other but not too far either. Let's say, I pick a 2 from 8 and give it to 15. I get two numbers 4 and 30. They are farther apart and their sum is 34. So the numbers we are looking for are -4 and -30 (to get -34 as sum)

Taking another example: 8x^2 - 47x - 63

a + b = -47
a*b = 8*(-63) = - 2*2*2*3*3*7

One of a and b is negative since the product is negative. So one of a and b is greater than 47.
I split the primes into two groups: 2*2*2 and 3*3*7 to get 8 and 63 but 63 - 8 = 55. We need to go lower than 55 so we need to get the numbers closer together. 63 is way greater than 8.
Take off a 3 from 63 and give it to 8 to get 21 and 24. Too close.
Rather take off 7 from 63 and give it to 8 to get 9 and 56. Now, 56 - 9 = 47 so you have your numbers as 56 and 9. The greater one has to be negative since the sum is negative so you split the middle term as: -56 and 9.

With a little bit of practice, the hardest questions can be easily solved...