\[\begin{aligned}
x_1 &= -\frac{b}{4a}+\frac{S}{2}+\frac{1}{2}\sqrt{-4S^2-2p+\frac{q}{S}} \\
x_2 &= -\frac{b}{4a}+\frac{S}{2}-\frac{1}{2}\sqrt{-4S^2-2p+\frac{q}{S}} \\
x_3 &= -\frac{b}{4a}-\frac{S}{2}+\frac{1}{2}\sqrt{-4S^2-2p-\frac{q}{S}} \\
x_4 &= -\frac{b}{4a}-\frac{S}{2}-\frac{1}{2}\sqrt{-4S^2-2p-\frac{q}{S}}
\end{aligned}\]
Variables
aquartic coefficient
bcubic coefficient
pdepressed quartic parameter
qdepressed quartic parameter
Sauxiliary solution term
x1first root
x2second root
x3third root
x4fourth root
Description
What is this formula?
This formula solves quartic equations of the form:
ax^4+bx^3+cx^2+dx+e=0
It uses Ferrari's method to calculate the four possible roots of a fourth-degree polynomial.
When to use it
Use this formula when solving fourth-degree polynomial equations in algebra and engineering problems.
Example
For:
x^4-5x²+4=0
The solutions are:
x=±1
x=±2
Applications
- Advanced algebra
- Engineering mathematics
- Control theory
- Physical modeling
- Numerical analysis