Relation Between Roots and Coefficients Under Scaling

Relation Between Roots and Coefficients Under Scaling

## Formula

If:

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

has roots $$\alpha,\beta$$, then multiplying the entire equation by any non-zero constant $$k$$:

$$kax^2+kbx+kc=0$$

does not change the roots.

## Conditions / Special Cases

The ratios remain unchanged:

$$-\frac{kb}{ka}=-\frac ba$$

$$\frac{kc}{ka}=\frac ca$$

## Usage

- Used to simplify quadratic equations without changing their roots.