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

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Solution

The correct option is C
8

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

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 n=8, 15,24,35,.......99
$\Rightarrow$ Number of different values = 8

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x^{2} + 2 x - n = 0

n \in N

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x^{2} + 2 x - n = 0

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[5, 100]

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