Mathematics - Quadratic Equation Question with Solution | TestHub

For real and distinct roots, the discriminant $D>0$. Here, $D=(-2(k-1){)}^2-4(1)(k+5)=4(k^2-2k+1)-4(k+5)=4(k^2-3k-4)$. We need $4(k^2-3k-4)>0$, which means $k^2-3k-4>0$. Factoring, $(k-4)(k+1)>0$. This inequality holds when $k<-1$ or $k>4$. So, $k\in (-\infty ,-1)\cup (4,\infty )$.