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quartic formula

The four roots $r_1, r_2, r_3, r_4$ of a quartic polynomial equation $x^4 + ax^3 + bx^2 + cx + d = 0$ are given by \begin{eqnarray*} r_1 & = & {\frac{-a}{4} - \frac{1}{2}{\sqrt{\frac{a^2}{4} - \frac{2b}{3} + \frac{2^{\frac{1}{3}}\left( b^2 - 3ac + 12d \right) } {3{\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4{\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} \right) }^{\frac{1}{3}}} + \left(\frac{{ 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4 {\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} }} {54}\right)^\frac{1}{3}}} - \frac{1}{2}{\sqrt{\frac{a^2}{2} - \frac{4b}{3} - \frac{2^{\frac{1}{3}}\left( b^2 - 3ac + 12d \right) } {3{\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4{\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} \right) }^{\frac{1}{3}}} - \left(\frac{{ 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4 {\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} }} {54}\right)^\frac{1}{3} - \frac{-a^3 + 4ab - 8c} {4{\sqrt{\frac{a^2}{4} - \frac{2b}{3} + \frac{2^{\frac{1}{3}} \left( b^2 - 3ac + 12d \right) }{3 {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4 {\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} \right) }^ {\frac{1}{3}}} + \left( \frac{{ 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4 {\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} } }{54}\right)^\frac{1}{3}}}}}}} \\ r_2 & = & {\frac{-a}{4} - \frac{1}{2}{\sqrt{\frac{a^2}{4} - \frac{2b}{3} + \frac{2^{\frac{1}{3}}\left( b^2 - 3ac + 12d \right) } {3{\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4{\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} \right) }^{\frac{1}{3}}} + \left( \frac{{ 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4 {\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} }} {54}\right)^\frac{1}{3}}} + \frac{1}{2}{\sqrt{\frac{a^2}{2} - \frac{4b}{3} - \frac{2^{\frac{1}{3}}\left( b^2 - 3ac + 12d \right) } {3{\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4{\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} \right) }^{\frac{1}{3}}} - \left( \frac{{ 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4 {\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} }} {54}\right)^\frac{1}{3} - \frac{-a^3 + 4ab - 8c} {4{\sqrt{\frac{a^2}{4} - \frac{2b}{3} + \frac{2^{\frac{1}{3}} \left( b^2 - 3ac + 12d \right) }{3 {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4 {\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} \right) }^ {\frac{1}{3}}} + \left( \frac{{ 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4 {\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} } }{54}\right)^\frac{1}{3}}}}}}} \\ r_3 & = & {\frac{-a}{4} + \frac{1}{2}{\sqrt{\frac{a^2}{4} - \frac{2b}{3} + \frac{2^{\frac{1}{3}}\left( b^2 - 3ac + 12d \right) } {3{\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4{\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} \right) }^{\frac{1}{3}}} + \left( \frac{{ 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4 {\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} }} {54}\right)^\frac{1}{3}}} - \frac{1}{2}{\sqrt{\frac{a^2}{2} - \frac{4b}{3} - \frac{2^{\frac{1}{3}}\left( b^2 - 3ac + 12d \right) } {3{\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4{\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} \right) }^{\frac{1}{3}}} - \left( \frac{{ 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4 {\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} }} {54}\right)^\frac{1}{3} + \frac{-a^3 + 4ab - 8c} {4{\sqrt{\frac{a^2}{4} - \frac{2b}{3} + \frac{2^{\frac{1}{3}} \left( b^2 - 3ac + 12d \right) }{3 {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4 {\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} \right) }^ {\frac{1}{3}}} + \left( \frac{{ 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4 {\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} } }{54}\right)^\frac{1}{3}}}}}}} \\ r_4 & = & {\frac{-a}{4} + \frac{1}{2}{\sqrt{\frac{a^2}{4} - \frac{2b}{3} + \frac{2^{\frac{1}{3}}\left( b^2 - 3ac + 12d \right) } {3{\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4{\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} \right) }^{\frac{1}{3}}} + \left( \frac{{ 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4 {\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} }} {54}\right)^\frac{1}{3}}} + \frac{1}{2}{\sqrt{\frac{a^2}{2} - \frac{4b}{3} - \frac{2^{\frac{1}{3}}\left( b^2 - 3ac + 12d \right) } {3{\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4{\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} \right) }^{\frac{1}{3}}} - \left( \frac{{ 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4 {\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} }} {54} \right)^\frac{1}{3} + \frac{-a^3 + 4ab - 8c} {4{\sqrt{\frac{a^2}{4} - \frac{2b}{3} + \frac{2^{\frac{1}{3}} \left( b^2 - 3ac + 12d \right) }{3 {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4 {\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} \right) }^ {\frac{1}{3}}} + \left( \frac{{ 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd + {\sqrt{-4 {\left( b^2 - 3ac + 12d \right) }^3 + {\left( 2b^3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} } }{54}\right)^\frac{1}{3}}}}}}} \end{eqnarray*} The formulas for the roots are much too unwieldy to be used for solving quartic equations by radicals, even with the help of a computer. A practical algorithm for solving quartic equations by radicals is given in the concluding paragraph of the Galois-theoretic derivation of the quartic formula.

quartic formula is owned by David Jao.

How to Cite This Entry

David Jao. "quartic formula" (version 3). PlanetMath.org. Freely available at http://planetmath.org/QuarticFormula.html.

Classification

AMS MSC:

12D10 (Field theory and polynomials :: Real and complex fields :: Polynomials: location of zeros (algebraic theorems))

Pending Errata and Addenda

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Discussion

Failure at solving a quartic equation by potato_jack on 2012-03-18 22:46:24

Good evening.

Using the formula, I have tried to solve the following quartic equation:

0=x^4-2x^3+4x^2-3x-10 (i.e: a=-2 , b=4 , c=-3 , d=-10), which has 2 real solutions and 2 complex ones.

However, when I came to the last "part" of the formula - the last fraction with the numerator of (-a^3+4ab-8c) - I was surprised to see that, after having applied the required calculations, the denominator of this fraction ended up being equal to 0, making any attempt to solve the equation using the formula vain.

Have I missed something important?

Please give feedback.

[ reply | up ]

Re: Failure at solving a quartic equation by steven gregory on 2012-03-19 14:31:34

computer solution by ayjara on 2007-04-25 10:04:37

Hi,

"The formulas for the roots are much too unwieldy to be used for solving quartic equations by radicals, even with the help of a computer."

I think we could compute the roots almost "easily" which this procedure:

%%%% begin MATLAB code %%%%
a3 = a;
a2 = b;
a1 = c;
a0 = d;

T1 = -a3/4;
T2 = a2^2 - 3*a3*a1 + 12*a0;
T3 = (2*a2^3 - 9*a3*a2*a1 + 27*a1^2 + 27*a3^2*a0 - 72*a2*a0)/2;
T4 = (-a3^3 + 4*a3*a2 - 8*a1)/32;
T5 = (3*a3^2 - 8*a2)/48;

R1 = sqrt(T3^2 - T2^3);
R2 = cubic_root(T3 + R1);
R3 = (1/12)*(T2/R2 + R2);
R4 = sqrt(T5 + R3);
R5 = 2*T5 - R3;
R6 = T4/R4;

r1 = T1 - R4 - sqrt(R5 - R6);
r2 = T1 - R4 + sqrt(R5 - R6);
r3 = T1 + R4 - sqrt(R5 + R6);
r4 = T1 + R4 + sqrt(R5 + R6);
%%%% end MATLAB code %%%

I does some tests and I think this code is correct.
My (not ease) job was to extract the Ts and Rs from your set of equations...

The cubic_root function must be

%%%% MATLAB code %%%
function v = cubic_root(u)
if imag(u) == 0 & real(u) < 0
v = -(abs(u)^(1/3));
else
v = u^(1/3);
end
%%%% end MATLAB code %%%

I think this may be helpful.

Regards,
Adalberto Ayjara Dornelles Filho
Caxias do Sul, Brasil