Quartic Equation Solver
Quartic Equation Solver
\[\begin{aligned}
x_1 &= -\frac{b}{4a}+\frac{S}{2}+\frac{1}{2}\sqrt{-4S^2-2p+\frac{q}{S}} \\
x_2 &= -\frac{b}{4a}+\frac{S}{2}-\frac{1}{2}\sqrt{-4S^2-2p+\frac{q}{S}} \\
x_3 &= -\frac{b}{4a}-\frac{S}{2}+\frac{1}{2}\sqrt{-4S^2-2p-\frac{q}{S}} \\
x_4 &= -\frac{b}{4a}-\frac{S}{2}-\frac{1}{2}\sqrt{-4S^2-2p-\frac{q}{S}}
\end{aligned}\]
Variables
a = quartic coefficient
b = cubic coefficient
p = depressed quartic parameter
q = depressed quartic parameter
S = auxiliary solution term
x1 = first root
x2 = second root
x3 = third root
x4 = fourth 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
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