Two quadratic equations a_1x^2 + b_1x + c_1 = 0 and a_2x^2 + b_2x + c_2 = 0 may share one or both roots.

One common root: Subtract the equations to find the common root: x = (c_1a_2 - c_2a_1)/(a_1b_2 - a_2b_1) (provided the denominator is non-zero). The condition for a common root is (c_1a_2 - c_2a_1)^2 = (a_1b_2 - a_2b_1)(b_1c_2 - b_2c_1). This is derived by substituting the common root back into either equation.

Both roots common: a_1/a_2 = b_1/b_2 = c_1/c_2 (proportional coefficients). This means the two equations are essentially the same.

Equation formation from roots: If alpha and beta are given roots, the quadratic is x^2 - (alpha+beta)x + alpha*beta = 0. If roots are obtained by transformation (like squaring, reciprocal, adding a constant), use Vieta's formulas to compute the new sum and product.

For equations with irrational or complex roots: If 2 + sqrt(3) is a root with rational coefficients, then 2 - sqrt(3) is automatically a root. Similarly, if 3 + 2i is a root with real coefficients, then 3 - 2i must be a root. This conjugate pairing principle reduces the work of finding the equation.

JEE technique: When two equations are given with a parameter and the condition "common root" is imposed, subtract the equations to isolate the common root in terms of the parameter, then substitute back to get the parameter value.