Two quadratics: $a_{1x}^2$ + $b_{1x}$ + $c_1$ = 0 and $a_{2x}^2$ + $b_{2x}$ + $c_2$ = 0.

One common root: Let alpha be the common root. Subtracting: (a_{1-a}_2)$alpha^2$ + (b_{1-b}_2)alpha + (c_{1-c}_2) = 0. Alternatively, the condition is: ($c_1$$a_2$ - $c_2$$a_1$)^2 = ($a_1$$b_2$ - $a_2$$b_1$)($b_1$$c_2$ - $b_2$$c_1$)

Both roots common: $a_1$/$a_2$ = $\frac{b_1}{b_2}$ = $\frac{c_1}{c_2}$ (proportional coefficients).

Practical method: From the two equations, eliminate the quadratic term (or constant term) to get a linear equation in alpha, solve for alpha, then verify.