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An Integer Root Problem (Posted on 2006-11-17) Difficulty: 2 of 5
Determine whether or not x²+7x-14(q²+1)=0 has any integer roots for integer q.

See The Solution Submitted by K Sengupta    
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Solution Solution maybe? | Comment 1 of 2

I'm not sure if I'm doing this right, but I do not think there are any integer roots if q is an integer.

Solving using the quadratic formula means the roots are

-7 + sqrt[49 + 56(q^2+1)] / 2

For this to be an integer, sqrt[49 + 56(q^2+1)] must be an odd integer.  This simplifies to sqrt[7(15+8q^2)] so (15+8q^2) must be 0 mod 7, and then 8q^2 must be 6 mod 7.  However, there are no integer solutions for q for which this is true.


  Posted by tomarken on 2006-11-17 10:08:01
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