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_1a_2 - c_2a_1)^2 = (a_1b_2 - a_2b_1)(b_1c_2 - b_2c_1)

Both roots common: a_1/a_2 = b_1/b_2 = 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.