Question

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

Answer 1

The roots of the equation are:
\(\hspace7ex \dfrac{-2 \pm \sqrt{4-4(-n)}}{2}\)
\(\hspace7ex =\dfrac{-2 \pm \sqrt{4(1+n)}}{2}\)
\(\hspace7ex =\dfrac{-2 \pm 2\sqrt{1+n}}{2}\)
\(\hspace7ex =-1 \pm \sqrt{1+n}\)

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$.