Question

Consider the equation $x^2-2x-n=0$, where $n\in N$ and $n\in [5,100]$. The total number of different values of $n$ so that the given equation has integral roots is

• A. 8
• B. 3
• C. 6
• D. 4

Answer 1

The correct option is A 8

Given $x^2-2x-n=0$

The given equation has integral roots.

The sum of the roots $=2$

Let the roots be $1-k,k+1$ , where $k$ is a positive integer.

Product of the roots $=1-k^2=-n$

$\Rightarrow k^2-1=n$

$n\in [5,100]$

$5\le n\le 100$

$6\le n+1\le 101$

$6\le k^2\le 101$

$\Rightarrow k^2=9,16,25,36,64,49,64,81,100$

For $k=3,4,5,6,7,8,9,10$

Hence, there will be 8 corresponding values of $n$.