Common Root of Two Quadratics

Common Root of Two Quadratics

## Definition / Concept

Two quadratic equations have a common root if at least one value satisfies both equations simultaneously.

## Formula

If:

$$a_1x^2+b_1x+c_1=0$$

and:

$$a_2x^2+b_2x+c_2=0$$

have a common root, then a suitable linear combination can eliminate the $x^2$ term:

$$a_2(a_1x^2+b_1x+c_1)-a_1(a_2x^2+b_2x+c_2)=0$$

giving:

$$(a_2b_1-a_1b_2)x+(a_2c_1-a_1c_2)=0$$

## Usage

- Used to find a common root or derive a condition for two quadratic equations to share a root.