Question

Determine the limits between which $n$ must lie in order that the equation
$2ax(ax+nc)+(n^2-2)c^2=0$ may have real roots.

Answer 1

Given equation is $2ax(ax+nc)+(n^2-2)c^2=0$

The equation can be written as $2a^2{x}^2+2ancx+(n^2-2)c^2=0$

Given that the equation have real roots

For it to have real roots, the discriminant must be greater than or equal to zero.

$\Rightarrow (2anc{)}^2-8(ac{)}^2(n^2-2)\ge 0$

$\Rightarrow 4\left(ac{)}^2\right(n^2-2n^2+4)\ge 0$

$\Rightarrow 4\left(ac{)}^2\right(4-n^2)\ge 0$

The above equation is always true if $4-n^2\ge 0$

$\Rightarrow n^2\le 4$

$\Rightarrow -2\le n\le 2$