How to prove that $\frac{\sqrt{5(160ab+8b+13)}-5}{20}$ is not an integer 
Prove that $\frac{\sqrt{5(160ab+8b+13)}-5}{20}$ is not an integer where $a$ and $b$ are positive integers.

One of the roots of the equation $10x^2+5x-1-(20a+1)b=0$ is $\frac{\sqrt{5(160ab+8b+13)}-5}{20}$. Wolframalpha says that this equation has no integer solution. How can I prove that $\frac{\sqrt{5(160ab+8b+13)}-5}{20}$ is not an integer?

Take $b=14$ and $a=3$. In this case, $\sqrt{5(160ab+8b+13)} = 185$. 
This gives $x=\frac{185-5}{20}=9$ as a valid root. So there are integer solutions for $x$ when $a$ and $b$ are positive integers. 
If Wolfram told you there were no integer solutions, either Wolfram is wrong, or perhaps you made a typo when entering your equation.
