Determine whether or not x²+7x-14(q²+1)=0
has any integer roots for integer q.
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