Two quadratic equations $a_1$*$x^2$ + $b_1$x + $c_1$ = 0 and $a_2$$x^2$ + $b_2$*x + $c_2$ = 0 may share one or both roots.

One common root: Subtract the equations to find the common root: x = $\frac{c_1*a_2 - c_2*a_1}{(a_1*b_2 - a_2*b_1)}$ (provided the denominator is non-zero). The condition for a common root 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$). This is derived by substituting the common root back into either equation.

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