SOLUTION: Help me figure this out...The product of two consecutive negative integers is 462. Determine the two integers. How do I set it up and how do I solve it?

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Question 232492: Help me figure this out...The product of two consecutive negative integers is 462. Determine the two integers. How do I set it up and how do I solve it?
Found 2 solutions by jsmallt9, Alan3354:
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
Each integers is 1 more than the one before it. So if
x = the first negative integer
then
x+1 = the second integer.

The problem tells us that their product is 462:
%28x%29%2A%28x%2B1%29+=+462
Now we solve this. Start by simplifying the left side:
x%5E2+%2Bx+=+462
Since this is a quadratic equation, we'll get one side equal to zero (by subtracting 462 from each side):
x%5E2+%2B+x+-+462+=+0
Now we factor (or use the quadratic formula). Factoring a trinomial like this (with a leading coefficient of 1) is simply a matter of finding the answer to the question, "What are the factors of the constant term which add up to the coefficient of the middle term?". In your equation this question becomes: "What are the factors of -462 that add up to 1?" Since -462 is such a large number it may have a lot of possible pairs of factors. But if we think about the situation we can find the right pair fairly quickly. Since -462 is negative it must have one positive and one negative factor. When you add a positive and a negative number, as I hope you know, it is like subtracting. And you use the sign of the larger number. From this we know
  • The positive number is the larger one (because we want them to add up to +1)
  • The numbers, disregarding the sign, are one apart from each other like 8 and -7 or 67 and -66.
So we're looking for two numbers like this whose product is -462. Since 20*20 = 400 we can start near there. So try 21*(-20), 22*(-21), etc. until you get -462. And you quickly find that 22*(-21) works! So our trinomial factors into:
%28x%2B22%29%28x-21%29+=+0
From the Zero Product Property we know that this or any product is equal to zero only if one of the factors is zero. So:
x%2B22+=+0 or x-21+=+0
Solving these we get:
x+=+-22 or x+=+21
Since we are looking only for negative integers, we must reject the x=21 solution. So our consecutive negative integers are -22 (x) and -21 (x+1).

Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Do it the easy way.
sqrt%28462%29 = 21+
The numbers are 21 and 22, or -21 and -22.