How to find the root of imaginary number and the number of roots of higher order equation? For example, if x ^ 3-4x ^ 2-3x + 2 = 0, how can we know that it has several imaginary solutions and several real solutions?

How to find the root of imaginary number and the number of roots of higher order equation? For example, if x ^ 3-4x ^ 2-3x + 2 = 0, how can we know that it has several imaginary solutions and several real solutions?

First of all, we need to clarify that for a general equation of higher order, it is very possible that we can not judge its roots, sometimes the solution roots of these equations can not be expressed by basic mathematical symbols, and even use the knowledge of group theory
But for a cubic equation, if you want to judge its root, it is OK. Here we give a discriminant for a special case
For a univariate cubic equation: x ^ 3 + PX + q = 0, when △ = (Q / 2) ^ 2 + (P / 3) ^ 3 > 0, there is a real root and a pair of conjugate imaginary roots;
When △ = (Q / 2) ^ 2 + (P / 3) ^ 3 = 0, there are three real roots, two of which are equal;
When △ = (Q / 2) ^ 2 + (P / 3) ^ 3