Question

The number of integral values that $k$ can take if $x^2+kx-k^2+5k>0$ for all $x\in R$, is

Answer 1

$\Rightarrow$ The given quadratic equation is $x^2+kx-k^2+5k>0$

$\Rightarrow$ Here, $a=1,b=k,c=-k^2+5k$

Since, $x$ belongs to real number.

$\Rightarrow$ $b^2-4ac=0$

$\Rightarrow$ $(k{)}^2-4(1\left)\right(-k^2+5k)=0$

$\Rightarrow$ $k^2+4k^2-20k=0$

$\Rightarrow$ $5k^2-20k=0$

$\Rightarrow$ $k(5k-20)=0$

$\Rightarrow$ $k=0$ and $5k-20=0$

$\therefore$ $k=0,4$