Condition that a quadratic equation is positive/negative on infinite range

Question

A similar exercise to Find parameter of quadratic equation such that it is positive/negative on a range: $f:\mathbb{R}\rightarrow\mathbb{R},~f(x)=(a+3)x^2-(a+4)x+a+4, a\in\mathbb{R}-\{3\},~f(x)>0~\forall~x\in(-\infty;0)$. The conditions I would put:

1. Dominant coefficient positive $\Rightarrow a+3>0$
2. $\Delta<0$ (so the function will be positive on $\mathbb{R}$) or : $\Delta\ge0$ and $f(0)\ge0$ and $x_{min}\ge0$ (so the function won't touch the x-axis in the interval)

Is this OK or is there a more simple and elegant solution like in the other post?

Thanks a lot!

Answer 1

If $a<-3$, then $a+3<0$, and therefore the set of those $x$ such that $f(x)>0$ is bounded, and so it cannot contain $(-\infty,0)$.

And if $a>-3$, then:

• if $f(x)$ has no roots, you always have $f(x)>0$;
• otherwise, the product of the roots is $\frac{a+4}{a+3}>0$, and therefore both roots have the same sign; and the sum of the roots is also $\frac{a+4}{a+3}$, and so both of them are greater than $0$. So, $f(x)>0$ when $x\in(-\infty,0)$.