Quartic equation Totally Explained

Everything about Quartic Equation totally explained

In mathematics, a quartic equation is one which can be expressed as a quartic function equalling zero. The general form of a quartic equation is ť

ax^4+bx^3+cx^2+dx+e=0 ,

where

a
e 0

.
The quartic is the highest order polynomial equation that can be solved by radicals in the general case (for example, one where the coefficients can take any value).

History

Quartic equations were first considered by Jaina Mathematicians in ancient India between 400 BC and 200 AD. See History of Cubic equation for more details.
Lodovico Ferrari is attributed with the discovery of the solution to the quartic in 1540, but since this solution, like all algebraic solutions of the quartic, requires the solution of a cubic to be found, it couldn't be published immediately.(External Link

) The solution of the quartic was published together with that of the cubic by Ferrari's mentor Gerolamo Cardano in the book Ars Magna (1545).
The proof that this was the highest order general polynomial for which such solutions could be found was first given in the Abel–Ruffini theorem in 1824, proving that all attempts at solving the higher order polynomials would be futile. The notes left by Évariste Galois in 1832 later led to the complete theory of the roots of polynomials, of which this theorem was one result.

Applications

Polynomials of high degrees often appear in problems involving optimization, and sometimes these polynomials happen to be quartics, but this is a coincidence.
Quartics often arise in computer graphics and during ray-tracing against surfaces such as quadrics or tori surfaces, which are the next level beyond the sphere and developable surfaces.(External Link

)
   Another frequent generator of quartics is the intersection of two ellipses.
   In Computer-aided manufacturing, the torus is a common shape associated with the endmill cutter. In order to calculate its location relative to a triangulated surface, the position of a horizontal torus on the Z-axis must be found where it's tangent to a fixed line, and this requires the solution of a general quartic equation to be calculated. Over 10% of the computational time in a CAM system can be consumed simply calculating the solution to millions of quartic equations.
   A program demonstrating various analytic solutions to the quartic was provided in Graphics Gems Book V.(External Link

) However, none of the three algorithms implemented are unconditionally stable. In an updated version of the paper(External Link

), which compares the 3 algorithms from the original paper and 2 others, it's demonstrated that computationally stable solutions exist only for 4 of the possible 16 sign combinations of the quartic coefficients.

Solving a quartic equation

Special cases

Consider a quartic equation expressed in the form

a_0x^4+a_1x^3+a_2x^2+a_3x+a_4=0

Degenerate case

If a₄ (the constant term) = 0, then one of the roots is x = 0, and the other roots can be found by dividing by x, and solving the resulting cubic equation, ť

a_0x^3+a_1x^2+a_2x+a_3=0. ,

Evident roots: 1 and -1 and -k

Call our quartic polynomial Q(x). Since 1 raised to any power is 1,

Q(1)=a_0+a_1+a_2+a_3+a_4

. Thus if

a_0+a_1+a_2+a_3+a_4=0

, Q(1)=0 and so x=1 is a root of Q(x). It can similarly be shown that if

a_0+a_2+a_4=a_1+a_3

, x=-1 is a root.
In either case the full quartic can then be divided by the factor (x-1) or (x+1) respectively yielding a new cubic polynomial, which can be solved to find the quartic's other roots.
If

a_1 = a_0 k

a_2 = 0

and

a_4= a_3 k

, then x= - k is a root of the equation. The full quartic can then be factorized this way:

a_0 x^4+ a_0 k x^3 + a_3 x + a_3 k = a_0 x^3 (x +k ) + a_3 (x+k) = (a_0 x^3 + a_3) (x+k) ,

.
If

a_1 = a_0 k

a_3 = a_2 k

and

a_4 = 0

, x=0 and x= -k are two known roots. Q(x)divided by

x (x+k)

is a quadratic equation.

Biquadratic equations

A quartic equation where a₃ and a₁ are equal to 0 takes the form
ť

a_0x^4+a_2x^2+a_4=0,!

and thus is a biquadratic equation, which is easy to solve: let

z=x^2

, so our equation turns to ť

a_0z^2+a_2z+a_4=0,!

which is a simple quadratic equation, whose solutions are easily found using the quadratic formula:
ť

z= over

then ť

F_1 F_2 = x^4 + cx^2 + dx + e,,(4)

We therefore can solve the quartic by solving for w and then solving for the roots of the two factors using the quadratic formula.

Further Information

Get more info on 'Quartic Equation'.

External Link Exchanges

Do you know how hard it is to get a link from a large encyclopaedia? Well we're different and will prove it. To get a link from us just add the following HTML to your site on a relevant page:

<a href="http://quartic_equation.totallyexplained.com">Quartic equation Totally Explained</a>

Then simply click through this link from your web page. Our crawlers will verify your link, extract the title of your web page and instantly add a link back to it. If you like you can remove the words Totally Explained and embed the link in article text.
As long as your link remains in place, we'll keep our link to you right here. Please play fair - our crawlers are watching. Your site must be closely related to this one's topic. Any kind of spamming, dubious practises or removing the link will result in your link from us being dropped and, potentially, your whole site being banned.