If $n$ is a integer which lies in $[5, 100]$ , then the number of integral roots of the equation $x^{2} + 2 x - n = 0$ is

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Solution

Given equation
$x^{2} + 2 x - n = 0 \Rightarrow (x + 1)^{2} = 1 + n \Rightarrow x = - 1 \pm \sqrt{1 + n}$
For integral roots $n + 1$ should be a perfect square.
Possible values of $n + 1$ is
${9, 16, 25, 36, 49, 64, 81, 100}$
When $n + 1 = 9$
$x = - 1 \pm 3 = - 4, 2$
Similarly every value of $n + 1$ will give $2$ integral roots.

Hence, the total number of integral solution is $16$ .

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If n is a integer which lies in [5,100], then the number of integral roots of the equation x2+2x−n=0 is

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If $n$ is a integer which lies in $[5, 100]$ , then the number of integral roots of the equation $x^{2} + 2 x - n = 0$ is