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if I have this equation:

$f(x)=ax^2-bx+c $

and $f(x) \geq 0$

is that's mean the Discriminant, $\Delta=b^2-4ac \leq 0$ ?

and why ?

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    Depends on the context. It's quite common to ask this kind of question: "When is $x^2-1\geq 0$?". – 2017-02-20
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    if it is true for all $x$ and $a>0$ then yes, the discriminant is less than zero – 2017-02-20

3 Answers 3

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1) If you have $f(x)=ax^2-bx+c$ and $f(x)\ge 0$ for all real values of $x$ then you must have $a>0$ and $\Delta \le 0$ (I'm assuming that $a,b,c \in \Bbb R$).

2) If $f(x)\ge 0$ is just a inequality and you are trying to find a solution for it then not necessarily you must have $\Delta \le 0$.

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Assuming $a,b,c$ are all real numbers, of course, the answer isn't too bad. If the discriminant were positive, by the quadratic formula, you've have two distinct real roots and $f(x)$ would be negative between them.

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This is a quadratic equation. The roots are find with the Pythagoras equation and, in that equation, the delta is in a square-root. If the delta is less than zero no real solutions to this equations exists meaning that the function $f$ do not touch the $x$-axis. If the delta is zero then touches at least once and then exists one real solution to $f(x) = 0$.