Quadratic Inequalities

Quadratic Inequalities

## Definition / Concept

A quadratic inequality is an inequality involving an expression of degree two.

## Formula

For:

$$ax^2+bx+c>0$$

or:

$$ax^2+bx+c<0$$

first determine the roots and then use the sign of $$a$$ to identify the required intervals.

## Conditions / Special Cases

If $$D<0$$:

- If $$a>0$$:

$$ax^2+bx+c>0\quad\forall x\in\mathbb R$$

- If $$a<0$$:

$$ax^2+bx+c<0\quad\forall x\in\mathbb R$$

If $$D=0$$:

$$ax^2+bx+c=a(x-\alpha)^2$$

so its sign is determined entirely by $$a$$, except at $x=\alpha$.

## Usage

- Used to solve inequalities involving quadratic expressions.