Question

For what least integral $k$ is the quadratic trinomial $(k-2)x^2+8x+k+4$ positive for all values of $x$?

Answer 1

For positive value $D>0$

so $b^2-4ac>0$

$8^2-4(k-2)(k+4)$>0

$64-4k^2-8k+32>0$

$4k^2+8k-96<0$

$k^2+2k-24<0$

$k^2+6k-4k-24<0$

$k(k+6)-4(k+6)<0$

$(k+6)(k-4)<0$

$k<-6andk<4$