Difference between revisions of "Quartic Equation"
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==Solving Quartic Equations== | ==Solving Quartic Equations== | ||
− | Look in the "TLDR" | + | Look in the "TLDR" subsections for the final result of each step. |
===Bringing it down to a depressed quartic=== | ===Bringing it down to a depressed quartic=== | ||
Start with the equation <math>ax^4 + bx^3 + cx^2 + dx + e = 0.</math> | Start with the equation <math>ax^4 + bx^3 + cx^2 + dx + e = 0.</math> |
Revision as of 23:22, 2 August 2022
A quartic equation is an algebraic equation of the form
These types of equations are extremely hard to solve; however, there are very clever methods for solving them by bringing it down to a cubic. I am going to list the simplest of the five.
Contents
Solving Quartic Equations
Look in the "TLDR" subsections for the final result of each step.
Bringing it down to a depressed quartic
Start with the equation
Divide both sides by a:
Now, convert to a depressed quartic by substituting
.
We now have:
Now we have a depressed quartic: where
,
and
.
TLDR?
The new depressed quartic is
where
,
and
.
Descartes' Solution
René Descartes thought of factoring the depressed quartic into two quadratics: . Expanding the right-hand side gives
, simplifying to
. Equating coefficients gives the following system of equations:
from which we derive and substitute this:
Now eliminate and
by doing the following:
Substitute to get
This can be solved via the cubic formula. After is obtained, we have
and can now solve for
,
and
:
Solve for s
Solve for t and v
We have the system of equations . We can obtain
and
. Similarly,
.
Now that both factors have been obtained, we can solve for by using the quadratic formula on each of the factors. The two solutions for the quadratics combined form the four solutions of the depressed quartic; subtract
to each of the solutions to obtain the solutions to the original quartic.
TLDR?
is a nonzero solution to the cubic
(or subtract the two equations to obtain
). The solutions to the depressed quartic are
subtract
from each of the roots to obtain the roots of the original quartic.
Roots Rewritten
The roots are (
s must be the same sign).
The Quartic Formula
Be prepared: This formula is really complicated.
We are going to keep using in the derivation; we are going to substitute them into the final formula.
So, we start with .
We factor it into two quadratics: .
We have obtained . With
being a solution to
,
, according to the cubic formula,
Already messy. Therefore,
Then we substitute each of the s for those rather large expressions and simplify.
https://www.quora.com/What-is-the-general-formula-for-quartic-equation