SOLUTION: One positive integer is 2 less than twice another. The sum of their squares is 1009. Find the integers

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Question 1031569: One positive integer is 2 less than twice another. The sum of their squares is 1009. Find the integers
Answer by Alan3354(69443) About Me  (Show Source):
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One positive integer is 2 less than twice another. The sum of their squares is 1009. Find the integers
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Use x and 2x-2
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x%5E2+%2B+%282x-2%29%5E2+=+1009
5x%5E2+-+8x+%2B+4+=+1009
5x%5E2+-+8x+-+1005+=+0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 5x%5E2%2B-8x%2B-1005+=+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=%28-8%29%5E2-4%2A5%2A-1005=20164.

Discriminant d=20164 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--8%2B-sqrt%28+20164+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-8%29%2Bsqrt%28+20164+%29%29%2F2%5C5+=+15
x%5B2%5D+=+%28-%28-8%29-sqrt%28+20164+%29%29%2F2%5C5+=+-13.4

Quadratic expression 5x%5E2%2B-8x%2B-1005 can be factored:
5x%5E2%2B-8x%2B-1005+=+%28x-15%29%2A%28x--13.4%29
Again, the answer is: 15, -13.4. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+5%2Ax%5E2%2B-8%2Ax%2B-1005+%29

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x = 15
--> 15 & 28