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The quadratic formula is a formula well-known to anyone who remembers their high-school algebra courses; the cubic formula is less well-known, but most mathematicians are probably at least glancingly familiar with it; the quartic formula, however, is generally not known except for being the largest degree of equation solvable with radicals. Although this is not the first or the most well-known method, the simplest way to solve the quartic is to split it into two quadratics.

The first step, however, is to convert the general quartic equation of the form x4+ax3+bx2+cx+d into a depressed quartic equation y4+py2+qy+r; this can be done by setting x=y+a/4 and substituting, much like completing the square in the quadratic equation and the depressed cubic that is the start of most solutions to the cubic formula. Next, set up the division of the depressed quartic into two quadratics:

y4+py2+qy+r=(y2+uy+v)(y2-uy+w)=y4+(v+w-u2)y2+(w-v)uy+vw

Note that the linear coefficients of the two quadratic terms are additive inverses of each other; this ensures that the cubic coefficient of their product is zero. Identifying the other coefficients gives us the following simultaneous equations for u, v, and w:

p=v+w-u2
q=(w-v)u
r=wv

The first two equations may be combined to define v and w in terms of u:

v=u2+p-q/u
w=u2+p+q/u

and an equation for u itself can be derived from these two and the third of the simultaneous equations:

r=wv=(u2+p)2-(q/u)2=u4+2pu2+p2-q2u-2

If one multiplies this equation by u2, one gets a cubic equation in the variable u2 which can be solved by the cubic formula. The three values of u2 (in the most general case) multiplied by the two values of u for each value of u2 corresponds to one of the 3 choose 2=6 possible quadratic polynomials which divide the quartic, which can be solved using the quadratic formula. The fact that all methods of solving the quartic use this or a closely related cubic equation can be related to the decomposition of the symmetric group of order 4 into the symmetric group of order 3 and the Klein four group; this analysis, when extended into other equations, is called Galois theory.

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