| Condition to Check | Result |
| If g(x) has one real root | f(x) may have one minimum at that point |
| If g(x) has three real roots | f(x) has two minimums and one maximum point |
| If α4 < 0 | f(x) has real roots |
| If α4 = 0 | f(x) has a real root at x = 0 |
| If α4 > 0 | f(x) may or may not have real roots |
| If g(x) has one real root repeated three times, and at that same point f(x) is also zero | that root is a real root of f(x) repeated four times |
| If g(x) has one real root and at that same point f(x) is also zero | that root is a repeated real root of f(x) |
| If f(x) is not zero or negative at any minimum point | the quartic equation has no real root |
| If g(x) has one real root and f(x) is zero or negative at the minimum point | f(x) has two real roots (or one repeated real root) and two complex conjugate roots |
| If g(x) has three real roots, and f(x) is negative at two minimum points and is not negative at the maximum point | all the roots of the quartic equation are real |