Question

Find the value(s) of $k$ for which the equation $x^2+5kx+16=0$ has no real roots.

Answer 1

The given equation is $x^2+5kx+16=0$

Here, $a=1,b=5k,c=16$

$\therefore D=b^2-4ac={\left(5k\right)}^2-4\times 1\times 16=25k^2-64$

The given equation will have no real roots, if
$D<0$

$\Rightarrow 25k^2-64<0\Rightarrow 25(k^2-\frac{64}{25})<0\Rightarrow k^2-\frac{64}{25}<0$ ($\because ab<0$ and $a>0\Rightarrow b<0$)
$\Rightarrow -\frac{8}{5}<k<\frac{8}{5}[\because x^2-a^2<0\Rightarrow -a<x<-a]$