Location of Roots Relative to a Point

Location of Roots Relative to a Point

## Formula

For:

$$f(x)=a(x-\alpha)(x-\beta),\qquad \alpha<\beta$$

the condition for both roots to lie to the right of $$k$$ is:

$$k<\alpha<\beta$$

The condition for both roots to lie to the left of $$k$$ is:

$$\alpha<\beta<k$$

## Useful Conditions

For both roots greater than $$k$$:

$$f(k)>0,\qquad \alpha+\beta>2k$$

when $a>0$.

For both roots less than $$k$$:

$$f(k)>0,\qquad \alpha+\beta<2k$$

when $a>0$.

## Usage

- Used in parameter problems asking for both roots to lie on a specified side of a number.