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Is there any way by which we can directly conclude whether a quadratic has integral roots or not?

Actually I was doing this question :

$$1 + 2 + 3 + 4 + ...... + n = kkk$$

Here I got $$n(1 + n)/2 = kkk$$

Since $kkk$ is always a multiple of $3$, so I put $kkk = 111$ and then checked if $$n(1 + n)/2 = 111$$ has an integral root or not.

Finally, I had to check till $kkk = 666$ which gave me $n = 36$

So, I want to know is there any quicker way by which I can just conclude by seeing if the quadratic has integral roots or not.

Sorry if my question is too vague or too trivial.

Please help.

Thanks.

  • 3
    You've seen the rational root theorem?2012-07-14
  • 0
    No sir, please provide me a link. Thanks.2012-07-14
  • 7
    I'm telling you to search for it yourself.2012-07-14
  • 1
    That happens when we have $$\frac{-b±\sqrt{b^2-4ac}}{2a}$$ as integer.2012-07-14
  • 0
    Well played @Nerd-Herd2012-07-14
  • 0
    Got something from an answer below?2012-08-11

2 Answers 2