the standard cubic equation ax^3+bx^2+cx+d=0 (a, b, c, d∈r, and a≠0) has the following solutions: 1. italian scholar cardan published it in 1545 the kaldan formula method; 2. the shengjin formula method published by chinese scholar fan shengjin in 1989.

both formula methods can solve standard cubic equations of one variable. due to the complexity of using cardan's formula to solve problems, in comparison, shengjin's formula is simple and clear, easy to remember, and the actual problem solving is more intuitive and more efficient.

shengjin formula identification method

when a=b=0, the equation has a triple real root.

when δ=b^2-4ac>0, the equation has one real root and a pair of conjugate imaginary roots.

when δ=b^2-4ac=0, the equation has three real roots, including a double root.

when δ=b^2-4ac<0, the equation has three unequal real roots.