Question

Consider the equation $x^2+2x-n=0$, where $n\in [5,100]$. Total number of different values of ${}^{\prime }{n}^{\prime }$ so that the given equation has integral roots is:

• A. $4$
• B. $3$
• C. $6$
• D. $8$

Answer 1

The correct option is D

$8$

The roots of the equation are:
$\frac{-2\pm \sqrt{4-4(-n)}}{2}$
$=\frac{-2\pm \sqrt{4(1+n)}}{2}$
$=\frac{-2\pm 2\sqrt{1+n}}{2}$
$=-1\pm \sqrt{1+n}$

It will be an integer when $\sqrt{1+n}$ is a perfect square. Given $n\in [5,100]$, $\sqrt{1+n}$ will be a perfect square when$1+n=9,16,25,36,49,64,81,100\Rightarrow n=8,15,24,35,.......99$
$\Rightarrow$ Number of different values of $n$ is $8$.