SOLUTION: The product of two consecutive integers is 89 more than their sum. Find the integers.

Question 614910: The product of two consecutive integers is 89 more than their sum. Find the integers.
Answer by ashipm01(26) (Show Source): You can put this solution on YOUR website!
The problem basically consists of two consecutive integers, as the problem statement mentions. Since both numbers are integers, one number can be denoted as , or whatever variable you want to use and then the other number would be since the numbers are consecutive.

Now that the numbers can be represented as and , the problem statement can be rewritten as something like:

The only thing left to do now is to solve for . To do that, lets start by simplifying the previous equation a little bit by multiplying out the left hand side of the equation and combining terms on the right hand side of the equation:

Next, you can move all terms on one side and then either factor the resulting equation for n, or you can use the quadratic formula to solve for n. Either way will work.

I will choose to factor this out as it will be faster than using the quadratic formula, but again the quadratic formula will work just fine as well.

To factor this, you need to find two numbers that multiply together to give -90 and add together to result in -1. If the numbers are not obvious at first, you can always list out the prime factors for -90 and try to find two of those prime factors that add together to give -1. For this problem, -10 and 9 will work. So now the problem can be factored as:

So from that, there are two solutions to this problem, n = 10 and n = -9. You can verify this by substituting both values back in the original equation:

When n = -9 :

Likewise, for n - 10:

So there are two solutions to this problem: {-9, -8} and {10, 11}
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