Q1. How to solve a Quartic Equation. There is an online calculator available (and many more similar) that gives the precise answers and also defines the method. Does anyone know what the source of this method is?
Q2. Given a Quartic Equation
$$
ax^4+bx^3+cx^2+dx+e=0\,,
$$
what are the conditions for the existence of real roots of the above equation?
Any reference material?
Best Answer
What about a set of expressions in the quartic's coefficients that discriminate between all cases? There are 9 cases:
4 distinct real roots.
3 distinct real roots with one of them being a double root.
2 distinct double roots both real.
Triple root and and a distinct fourth root.
Quadruple root.
2 distinct real roots and two complex roots.
double real root and 2 complex roots.
2 double roots both complex.
four distinct complex roots.
Such sets are known for quadratic and cubic polynomials:
Quadratic ($ ax^2+bx+c $):
Discriminant: $b^2-4ac$
Positive for two distinct real roots, zero for double root, and negative for complex conjugate roots.
Cubic ($ ax^3+bx^2+cx+d $):
$\Delta_1=2b^3-9abc+27a^2d$ and $\Delta_2=\Delta_1^2-4(b^2-3ac)^3$.
Then:
$ \Delta_2>0 $ gives one real root and two complex roots.
$ \Delta_2<0$ gives three distinct real roots.
$ \Delta_2=0 $ but $ \Delta_1\neq 0$ gives a double root plus one different root.
$ \Delta_1=\Delta_2=0$ gives a triple root.