Write the cubic in standard form: (ax^3+bx^2+cx+d=0)
If possible, factor out a common factor first
Test rational roots using the Rational Root Theorem
Substitute any rational root (r) into the polynomial
If (r) is a root, divide the cubic by ((x-r))
Solve the resulting quadratic equation
Use the quadratic formula if needed
If no rational roots are found, depress the cubic with (x=t-frac{b}{3a})
Rewrite it as (t^3+pt+q=0)
Compute (p) and (q)
Use Cardano’s formula:
(t=sqrt[3]{-frac{q}{2}+sqrt{left(frac{q}{2}right)^2+left(frac{p}{3}right)^3}}+sqrt[3]{-frac{q}{2}-sqrt{left(frac{q}{2}right)^2+left(frac{p}{3}right)^3}})
Convert back using (x=t-frac{b}{3a})
If the discriminant is negative, use trigonometric or complex methods
Verify all solutions by substitution into the original equation