Consider the equation $x^2+4 x-n=0$, where $n \in[20,100]$ is a natural number. Then the number of all distinct values of $n$, for which the given equation has integral roots, is equal to
Solution
<p>$$\begin{aligned}
& x^2+4 x-n=0 \text { has integer roots } \\
& \Rightarrow x=\frac{-4 \pm \sqrt{16+4 n}}{2}=-2 \pm \sqrt{4+n}
\end{aligned}$$</p>
<p>For $x$ to be integer $4+n$ must be perfect squares</p>
<p>$$\begin{aligned}
& n \in[20,100] \\
& n+4 \in[24,104]=S
\end{aligned}$$</p>
<p>$\left\{25,36, \ldots 10^2\right\} \in S \Rightarrow 5^2, 6^2, \ldots 10^2 \Rightarrow 6$ values of $n$</p>
About this question
Subject: Mathematics · Chapter: Complex Numbers and Quadratic Equations · Topic: Complex Numbers and Argand Plane
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