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

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answer is C.

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Detailed Solution

The roots of $x^{2} + 2 x - n = 0 are \frac{- 2 \pm \sqrt{4 + 4 n}}{1}$ which are integers $\Rightarrow \sqrt{4 + 4 n} = 2 \sqrt{1 + n}$ is an integer and $n \in [5, 100]$

$\Rightarrow n = 8, 15, 24, 35, 48, 63, 80, 99$

so number of equations = 8.

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