If a root of the equations ${\mathrm{x}}^2+\mathrm{px}+\mathrm{q}=0$ and ${\mathrm{x}}^2+\mathrm{ax}+\mathrm{\beta }=0$ is common, then its value will be (where $\mathrm{p}\ne \mathrm{\alpha } \mathrm{and} \mathrm{q}\ne \mathrm{\beta })$

Let the common root be y. Then ${\mathrm{y}}^2+\mathrm{py}+\mathrm{q}=0$ and $y^2+\alpha y+\beta =0$ 

on solving by cross multiplication, we have

$\frac{{\mathrm{y}}^2}{\mathrm{p\beta }-\mathrm{q\alpha }}=\frac{\mathrm{y}}{\mathrm{q}-\mathrm{\beta }}=\frac{1}{\mathrm{\alpha }-\mathrm{p}}$

$\therefore \mathrm{y}=\frac{\mathrm{q}-\mathbf{\beta }}{\mathrm{\alpha }-\mathrm{p}} \mathrm{and} \frac{{\mathrm{y}}^2}{\mathrm{y}}=\mathrm{y}=\frac{\mathrm{p\beta }-\mathrm{q\alpha }}{\mathrm{q}-\mathrm{\beta }}$