Difference between revisions of "Quartic Equation"
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− | A quartic equation is an algebraic equation of the form | + | A quartic equation is an algebraic [[equation]] of the form |
<math>ax^4 + bx^3 + cx^2 + dx + e = 0.</math> | <math>ax^4 + bx^3 + cx^2 + dx + e = 0.</math> |
Revision as of 19:55, 10 December 2020
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" section 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
.
You now have:
Now you 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
The solutions to the depressed quartic are
subtract
from each of the roots to obtain the roots of the original quartic.