Roots in an Interval

Roots in an Interval

## Definition / Concept

Conditions involving the values of a quadratic at the endpoints, together with the vertex, can determine whether its roots lie inside a given interval.

## Formula

For roots $$\alpha,\beta$$ to lie in $(p,q)$:

$$p<\alpha<\beta<q$$

Necessary conditions include:

$$f(p)\text{ and }f(q)\text{ having the same sign as }a$$

and:

$$-\frac{b}{2a}\in(p,q)$$

## Usage

- Used to locate both roots inside a specified interval.