Question

Total number of values of $a$ so that $x^2-x-a=0$ has integral roots, where $a\in N$ and $6\le a\le 100$, is equal to

Answer 1

$x^2-x-a=0$
$D=1+4a$ which is odd.
So, for integral roots, $D$ must be a perfect square for some odd integer.
Let $D=(2k+1{)}^2$ where $k=0,1,2,...$
$\Rightarrow 1+4a=4k^2+4k+1$
$\Rightarrow a=k(k+1)$
Since, $6\le a\le 100\Rightarrow 2\le k\le 9$
Therefore, possible values of $a$ are $6,12,20,30,42,56,72,90$