SOLUTION: The product of two consecutive integers is 9 less than 9 times their sum. Find the integers.

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Question 112422: The product of two consecutive integers is 9 less than 9 times their sum. Find the integers.
Answer by solver91311(24713) About Me  (Show Source):
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If the smaller of the two consecutive integers is x, then the other one must be x + 1.

The product of these two integers is then: x(x+1), and the sum of these two integers is x + (x + 1), or 2x + 1.

Now we can translate the first sentence of the problem into mathematical symbols:

The product of two consecutive integers (x(x + 1)) is (=) 9 times their sum (9(2x + 1)), less 9 (-9). Or,

x%28x%2B1%29=%289%282x%2B1%29%29-9

Now simplify and solve:

x%5E2%2Bx=18x%2B9-9
x%5E2%2Bx-18x=0
x%5E2-17x=0
x%28x-17%29=0, and finally,
x=0 or x=17.

This means that the two consecutive integers are either 0 and 1 or 17 and 18.

Check:
0%2A%280%2B1%29=9%2A%282%2A0%2B1%29-9
0=0, and our first answer checks.

17%2A18=9%2A%2817%2B18%29-9
306=9%2A%2835%29-9
306=315-9=306, and the second answer checks also.