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  4. The sum of all possible values of n where n ∈ N , x >0 and 10< n ≤ 100 such that the equation [2 x 2]+ x - n =0 has a solution, is equal to [Note: [ x ] denotes largest integer less than or equal to x.]

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Q. The sum of all possible values of $n$ where $n \in N , x >0$ and $10< n \leq 100$ such that the equation $\left[2 x ^2\right]+ x - n =0$ has a solution, is equal to
[Note: $[ x ]$ denotes largest integer less than or equal to $x$.]

Relations and Functions - Part 2

Solution:

We have $\left[2 x^2\right]+x-n=0$
$\Rightarrow x \text { has to be an integer. }$
$\Rightarrow n =2 x ^2+ x = x (2 x +1) $
$\therefore $ n can be $21,36,55,78$ corresponding to $x=3,4,5,6$.
Hence, sum of all possible values of $n$ is equal to 190. Ans.
Note: If $x$ is negative also then answer is 435 .]