Roots 2

Let $\alpha$ and $\beta$ , and $\alpha$ and $\gamma$ be the two roots of $x^2+px+q=0$ and $x^2+bx+a=0$ respectively. Therefore the common root is $\alpha$ .

By Vieta's formula , we have:

$\begin{cases} \alpha +\beta = -p &…(1) \\ \alpha \beta = q &…(2) \\ \alpha +\gamma = -b &…(3) \\ \alpha \gamma = a &…(4) \end{cases}$

$\begin{aligned} (1)-(3): \ \beta - \gamma & = b-p &…(5) \end{aligned}$

$\begin{aligned} \frac{(2)}{(4)}: \ \frac \beta \gamma & = \frac qa \\ \implies \gamma & = \frac aq \beta &…(6) \end{aligned}$

$\begin{aligned} \implies (5): \ \beta - \frac aq \beta & = b-p \\ \implies \beta & = \frac {qb-qp}{q-a} &…(7) \quad \small \color{#3D99F6} \text{Note that }q-a \ne 0 \text{ for } \beta \ne \gamma \end{aligned}$

$\begin{aligned} \implies (1):\ \alpha +\frac {qb-qp}{q-a} & = -p \\ \implies \alpha & = \boxed {\dfrac {pa-qb}{q-a}} \end{aligned}$