Question

If $n$ is a positive integer and $n\in [5,100]$, then the number of integral roots of the equation $x^2+2x-n=0$ are:

• A. $4$
• B. $16$
• C. $8$
• D. $10$

Answer 1

The correct option is B $16$

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

$\Rightarrow x^2+2x+1-1-n=0$

$\Rightarrow (x+1{)}^2=n+1$

$\Rightarrow x=-1\pm \sqrt{1+n}$ ..... (1)

$x$ will be an integer when $1+n$ is a perfect square.

$\Rightarrow 1+n=k^2$.

We have,

$5\le n\le 100$

$\Rightarrow 6\le n+1\le 101$

$\Rightarrow 6\le k^2\le 101$

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

$\Rightarrow$ $n+1$ has 8 different possible values.

$\therefore$ Possible values of $x$ by putting $n+1$ values in eq. (1) we get,

$x=-1\pm 3,-1\pm 4,-1\pm 5,-1\pm 6,-1\pm 7,-1\pm 8,-1\pm 9,-1\pm 10$

$x=-4,2,-5,3,-6,4,-7,5,-8,6,-9,7,-10,8,-11,9$

Hence, there are 16 possible integral solutions to the given equation.