<s>[INST] Show that the largest root of the following equation is $Show that the largest root of the following equation is $<n;n>2$.
$x^2-(n-2)x-2(n-2)$.
On solving I got the largest root $a=\frac{(n-2)+\sqrt {(n-2)^2+8(n-2)}}{2}$ but unable to show how is it $<n$.
I tried for various values of $n$ where the result is holding .But to prove it.
Please help
 [/INST] Assuming $n \ge 2$ . . .

One can just simplify the inequality $a < n$ until it admits to being true:
\begin{align*}
&a < n\\[5pt]
\iff\;&\frac{(n-2)+\sqrt {(n-2)^2+8(n-2)}}{2} < n\\[6pt]
\iff\;&(n-2) + \sqrt {(n-2)^2+8(n-2)} < 2n\\[6pt]
\iff\;&\sqrt {(n-2)^2+8(n-2)} < n+2\\[6pt]
\iff\;&(n-2)^2+8(n-2) < (n+2)^2\qquad\text{[since $n \ge 2$]}\\[5pt]
\iff\;&(n^2 - 4n + 4) + (8n-16) < n^2 + 4n + 4\\[5pt]
\iff\;&n^2 + 4n -12< n^2 + 4n + 4\\[5pt]
\iff\;&-12 < 4\\[5pt]
\end{align*}
which is clearly true.
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