Curves lying above or below the x-axis.

Question

The curve $y = (k+1)x^2 - 3x + (k+1)$ lies below the x-axis. Find a set of values for $k$. So, I know this means there are no real solutions for $y = 0$, hence we use the $b^2 - 4ac < 0$ method. However, why is the answer just $k<-5/2$ and not $k<-5/2$ OR $k>1/2$?

Answer 1

Because if the discriminant is negative it just means no real solutions. It does not guarantee that the function is all negative or all positive, just that it does not change sign. So you need to plug in some number, and see the value of $y$. The trivial one is $x=0$.

Answer 2

it must be $k+1<0$ and $$(k+1)\frac{9}{4(k+1)^2}-\frac{9}{2(k+1)}+k+1<0$$ can you solve this?