Common Roots of Two Quadratics

Common Roots of Two Quadratics

## Formula

For:

$$f(x)=a_1x^2+b_1x+c_1$$

and:

$$g(x)=a_2x^2+b_2x+c_2$$

a common root $\alpha$ satisfies:

$$f(\alpha)=g(\alpha)=0$$

A necessary condition for a common root, when the quadratics are not proportional, is obtained from:

$$\alpha=-\frac{a_2c_1-a_1c_2}{a_2b_1-a_1b_2}$$

provided:

$$a_2b_1-a_1b_2\ne0$$

## Usage

- Used to obtain the common root directly when the quadratic terms are eliminated.