Conditions for a quadratic equation to be positive?

Question

I was solving a problem in physics when I encountered this problem. I wanted to find the condition where the value of a quadratic equation is positive. Suppose the function is : $f(x) = Ax^2+Bx+C$
Then, I need to find the condition for which the function will give me positive value.Please Help!

Answer 1

We can express

$$y = ax^2 + bx +c= \frac{(2ax+b)^2 +4ac - b^2}{4a}$$

Now it is clear, $y$ is positive when $4a$ is positive and $4ac - b^2$ is positive.

Answer 2

Let $x_1,x_2$ be the solutions of $f(x)=0$.

Case 1: $x_1,x_2 \notin \mathbb R$. If $A>0$, then $f(x) >0$ for all $x \in \mathbb R.$ If $A<0$, then $f(x) <0$ for all $x \in \mathbb R.$

Case 2: $x_1,x_2 \in \mathbb R$. We can assume that $x_1 \le x_2$. If $A>0$ , then $f(x)>0$ $ \iff x<x_1$ or $x>x_2$. It is now your turn to investigate the situation if $A<0$.