How many real numbers satisfy x + x − 1 so that it is a Whole number.I came across this in a discrete mathematics book, and i wonder if my solution is correct.From looking at y = x + x − 1 I see only 2 possibilities, x ={1;-1;}Was this just so easy or am I missing something?

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How many real numbers satisfy   x  +      x          −      1       so that it is a Whole number.I came across this in a discrete mathematics book, and i wonder if my solution is correct.From looking at   y  =  x  +      x          −      1      I see only 2 possibilities, x ={1;-1;}Was this just so easy or am I missing something?

domino671v · Accepted Answer

For any integer n we want  n  =  x  +      1    x      ⟺        x    2    −  n  x  +  1  =  0We need   Δ  =      n    2    −  4  ≥  0 for the above equation to have solutions, so       |    n      |    ≥  2And the solutions are   x  =                    n        ±                              n            2                    −          4                    2       (note that       x          −      1        =                    n        ∓                              n            2                    −          4                    2       in that case).For   n  =  2 you get the   (  1  ,  1  ) solutionFor   n  =  −  2 you get   (  −  1  ,  −  1  ) solutionFor   n  =  4 you get the nice   (  2  +      3    ,  2  −      3    ) solutionand so on ...

Aidyn Crosby · Accepted Answer

Guide:Let  x  +      1    x    =  k      x    2    +  1  =  k  xThis is a quadratic equation. For which values of k does this has a real solution?

How many real numbers satisfy x+x^(−1) so that it is a Whole number. I came across this in a discrete mathematics book, and i wonder if my solution is correct. From looking at y=x+x^(−1) I see only 2 possibilities, x ={1;-1;} Was this just so easy or am I missing something

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