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quartic formula (Theorem)

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

$\displaystyle r_1$ $\displaystyle =$ $\displaystyle {\frac{-a}{4} - \frac{1}{2}{\sqrt{\frac{a^2}{4} - \frac{2b}{3} + ... ...3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} } }{54}\right)^\frac{1}{3}}}}}}}$  
$\displaystyle r_2$ $\displaystyle =$ $\displaystyle {\frac{-a}{4} - \frac{1}{2}{\sqrt{\frac{a^2}{4} - \frac{2b}{3} + ... ...3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} } }{54}\right)^\frac{1}{3}}}}}}}$  
$\displaystyle r_3$ $\displaystyle =$ $\displaystyle {\frac{-a}{4} + \frac{1}{2}{\sqrt{\frac{a^2}{4} - \frac{2b}{3} + ... ...3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} } }{54}\right)^\frac{1}{3}}}}}}}$  
$\displaystyle r_4$ $\displaystyle =$ $\displaystyle {\frac{-a}{4} + \frac{1}{2}{\sqrt{\frac{a^2}{4} - \frac{2b}{3} + ... ...3 - 9abc + 27c^2 + 27a^2d - 72bd \right) }^2}} } }{54}\right)^\frac{1}{3}}}}}}}$  

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 djao.
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See Also: Galois-theoretic derivation of the cubic formula, cubic formula, Cardano's derivation of the cubic formula, fundamental theorem of Galois theory

Other names:  biquadratic formula, quartic equation, biquadratic equation

Attachments:
Galois-theoretic derivation of the quartic formula (Proof) by djao
Ferrari-Cardano derivation of the quartic formula (Proof) by djao
Euler's derivation of the quartic formula (Theorem) by pahio
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Cross-references: Galois-theoretic derivation of the quartic formula, algorithm, even, radicals, equation, polynomial, roots
There are 8 references to this entry.

This is version 3 of quartic formula, born on 2002-01-22, modified 2005-07-07.
Object id is 1525, canonical name is QuarticFormula.
Accessed 85261 times total.

Classification:
AMS MSC: 12D10 (Field theory and polynomials :: Real and complex fields :: Polynomials: location of zeros )
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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
 
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