A cubic equation is an algebraic equation of the third degree of the form
a(x)=0,
where
a(x)=a0x3 + a1x2 + a2x + a3,
the coefficients
ai – are real (or complex) numbers, and
a0 ≠ 0.
According to the fundamental theorem of algebra, a cubic equation always has 3 roots (taking into account multiplicity). For an equation with real coefficients, if the root is a complex number, then the complex conjugate will also be its root. Therefore, at least one of the roots of an equation with real coefficients is always real.
Thus, for a cubic equation with real coefficients, the following combinations of three roots are possible:
1) three different real roots;
2) one real root and two complex conjugates;
3) two coinciding real roots (i.e. a root of multiplicity 2) and one that is different from these two real roots;
4) three coinciding real roots (i.e. a root of multiplicity 3).
Methods for solving cubic equations
Solving symmetric cubic equations
Solving cubic equations by factorization
Cardano's formula for solving cubic equations
