Is there any condition based on the coefficients of terms that guarantees all real solutions to a general cubic polynomial? e.g. $$ax^3+bx^2+cx+d=0\, ?$$
If not, are there methods rather than explicit formula to determine it?
Wikipedia says it's the following expression: $$ 18abcd-4b^3d + b^2c^2 -4ac^3 -27a^2d^2 $$ If it's positive there are three real roots. If it's negative there's one real and two (conjugate) imaginary roots. If it's equal to zero, there are fewer than three distinct roots, but they are all real.
Discriminant, $D = 18abcd - 4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2$
If $D < 0$, then the polynomial has two complex roots,
If $D = 0$, then there are three real roots, and two of them are definitely equal,
If $D > 0$, then there are three distinct real roots.
Refer http://en.wikipedia.org/wiki/Casus_irreducibilis for more information.