Quadratic Expression Always Negative

Quadratic Expression Always Negative

## Formula

For:

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

to be negative for every real $x$:

$$a<0$$

and:

$$D<0$$

For non-positive values:

$$f(x)\le0\quad\forall x\in\mathbb R$$

requires:

$$a<0,\qquad D\le0$$

## Usage

- Used when a quadratic inequality must hold for all real values of the variable.