# Common Roots of Quadratic Equations

In this section, we shall obtain conditions under which two given quadratic equations may have common roots.

## One Root Common

Let two quadratic equations be $\begin{array}{l} & ax^2+bx+c &= 0 \\ \text{and,} & a’x^2+b’x+c’ &= 0 \end{array}$

Let $\alpha$ be a root common to both the equations. Then, $\begin{array}{l} & a\alpha^2+b\alpha+c &= 0 \\ \text{and,} & a’\alpha^2+b’\alpha+c &= 0 \end{array}$

By the rule of cross multiplication, $\frac{\alpha^2}{bc’-b’c}=\frac{\alpha}{ca’-c’a}=\frac{1}{ab’-a’b}$ $\therefore\alpha=\frac{bc’-b’c}{ca’-c’a}=\frac{ca’-c’a}{ab’-a’b}$ $\therefore\frac{bc’-b’c}{ca’-c’a}=\frac{ca’-c’a}{ab’-a’b}$ $\text{or,}\;\;\;(bc’-b’c)(ab’-a’b)=(ca’-c’a)^2$ This is the required condition and the common root is $\frac{bc’-b’c}{ca’-c’a}\;\;\;\text{or,}\;\;\;\frac{ca’-c’a}{ab’-a’b}$

## Both Roots Common

Let $\alpha$ and $\beta$ be the common roots of the quadratic equations $\begin{array}{l} & ax^2+bx+c &= 0 \\ \text{and,} & a’x^2+b’x+c’ &= 0 \end{array}$

Then, $\alpha+\beta=-\frac{b}{a}=-\frac{b’}{a’}$ $\therefore\frac{a}{a’}=\frac{b}{b’}$ [From: Roots and Coefficients of a Quadratic Equation]

Again, $\alpha\beta=\frac{c}{a}=\frac{c’}{a’}$ $\therefore\frac{a}{a’}=\frac{c}{c’}$

Hence, $\frac{a}{a’}=\frac{b}{b’}=\frac{c}{c’}$ This is the required condition for the quadratic equations to have both roots common.