SOLUTION: when the square of three times a positive integer is decreased by the integer, the result is 2010. what is the integer?

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Question 530727: when the square of three times a positive integer is decreased by the integer, the result is 2010. what is the integer?
Answer by oberobic(2304) About Me  (Show Source):
You can put this solution on YOUR website!
x = a positive integer
3x = 3 times the number
9x^2 = the square of 3 times the number
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9x^2-x = 2010
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9x^2 -x -2010 = 0
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Using the quadratic equation, we find:
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Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 9x%5E2%2B-1x%2B-2010+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-1%29%5E2-4%2A9%2A-2010=72361.

Discriminant d=72361 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--1%2B-sqrt%28+72361+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-1%29%2Bsqrt%28+72361+%29%29%2F2%5C9+=+15
x%5B2%5D+=+%28-%28-1%29-sqrt%28+72361+%29%29%2F2%5C9+=+-14.8888888888889

Quadratic expression 9x%5E2%2B-1x%2B-2010 can be factored:
9x%5E2%2B-1x%2B-2010+=+9%28x-15%29%2A%28x--14.8888888888889%29
Again, the answer is: 15, -14.8888888888889. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+9%2Ax%5E2%2B-1%2Ax%2B-2010+%29

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So x = 15 is the only integer solution.
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Checking this value, we find:
9*225 -15 = 2010
correct.
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Answer: x = 15.