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Can anyone kindly tell me if there is a method (other than trial and error) to solve equations of the form below:

$$x^2 + x - 35 - 35[(x^2)/35] = 0$$

where $x$ is an integer and $[y]$ denotes the integer nearest to $y$?

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    Let's see... we have $[b]=\left\lfloor b+\frac12\right\rfloor$, and $p\bmod q=p-q\left\lfloor\frac{p}{q}\right\rfloor$...2011-11-08
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    A heuristic is to just remove the floor. Then we are looking at $$x^2+x-35-35\left(\frac{x^2}{35}\right)=x-35.$$ This is zero precisely when $x=35$, and we then expect that the solution of the original will be _close_ to this since it is only off by $35$ whereas $x^2$ has size 1225. Then plugging in 35, we see that $x=35$ is a solution of the original equation.2011-11-08
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    Apparently an endash rather than a hyphen was used as a minus sign within TeX. That makes sense considering that an actual typeset minus sign looks much longer than a stubby little hyphen. But when done in TeX, at least with mathJax or whatever it is they use here, it causes the minus sign not to have the usual spacing before it and after it. Strange.....2011-11-08
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    Anonymous user who keeps trying to insert "$x$ is an integer": Are you SPSmith? If so, please log in to make your edit.2011-11-09

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