If the equations $ax^{3}+3bx^{2}+3cx+d=0$ and $ax^{2}+2bx+c=0$ have a common root, then
- $\left(bc-ad\right)^{2}=3\left(ac-b^{2}\right)(bd+c^{2})$
- $\left(bc-ad\right)^{2}=2\left(ac-b^{2}\right)(bd+c^{2})$
- $\left(bc-ad\right)^{2}=4\left(ac+b^{2}\right)(bd+c^{2})$
- $\left(bc-ad\right)^{2}=4\left(ac-b^{2}\right)(bd-c^{2})$
This is a standard condition for a cubic and its derivative to share a root, derived from the resultant of the two polynomials.
Let the common root of the quadratic equation ax^2 + 2bx + c = 0 and the cubic equation ax^3 + 3bx^2 + 3cx + d = 0 be denoted as alpha. From the quadratic equation, we can express the relationship c = -a*alpha^2 - 2b*alpha. Substituting this expression for c into the cubic equation yields a*alpha^3 + 3b*alpha^2 + 3(-a*alpha^2 - 2b*alpha)*alpha + d = 0. Expanding the terms gives a*alpha^3 + 3b*alpha^2 - 3a*alpha^3 - 6b*alpha^2 + d = 0. Combining like terms results in the simplified equation -2a*alpha^3 - 3b*alpha^2 + d = 0. To eliminate alpha, we use the sum and product of the roots from the quadratic equation, where the sum of the roots is -2b/a and the product is c/a. By finding the sum of the cubes of the roots, alpha^3 + beta^3, and substituting the necessary identities, the final algebraic elimination yields (bc - ad)^2 = 4(ac - b^2)(bd - c^2).