Finding the smallest positive integers that satisfies given equations [closed]

Is it possible to find the smallest positive integer/s that satisfy a given equation or some inequality?

Example: $2x^2-3x>24$

Is there a formula for this?

closed as too broad by Aqua, Guy Fsone, mechanodroid, Arnaud D., kingW3Oct 22 '17 at 14:41

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• Start by checking if there is some small solution $a,b$. Then there are only finitely many possibilities to check for a smaller $a+b$. – hardmath Oct 18 '17 at 10:52

So $5|a(a-1)$ and since $a$ and $a-1$ are relatively prime $5$ divides only one of them, say $a$. Since $a_{\min}=5$ you get $b = 2$ and so $(a+b)_{\min} = 7$.
If $5$ divides $a-1$ then $a_{\min} = 6$ which is not good.
This is a very broad question. As for your specific equation, it's equivalent to $a(a-1) = 10b$ so you're basically looking for the smallest pair of successive numbers whose product is a multiple of 10. You also know that you need $a\geq5$ otherwise 5 can't divide $a(a-1)$, so you start looking at 5:
$5\times(5-1)=20=2\times10$. The smallest positive integers that satisfy your equation are $a=5$ and $b=2$.